EXISTENCE OF RADIAL SOLUTIONS P{LAPLACIAN ELLIPTIC ...

Manuscirpt submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume 15, Number 2, June 2006 pp. 447–479

EXISTENCE OF RADIAL SOLUTIONSFOR THE P–LAPLACIAN ELLIPTIC EQUATIONS WITH WEIGHTS

Elisa Calzolari, Roberta Filippucci and Patrizia Pucci

Dipartimento di Matematica e InformaticaUniversita degli Studi di Perugia

Via Vanvitelli 106123 Perugia, Italy

(Communicated by Eiji Yanagida)

Abstract. Using the definition of solution and the qualitative properties es-tablished in the recent paper [17], some existence results are obtained both forcrossing radial solutions and for positive or compactly supported radial groundstates in Rn of quasilinear singular or degenerate elliptic equations with weightsand with non–linearities which can be possibly singular at x = 0 and u = 0,respectively. The technique used is based on the papers [1] and [12]. Fur-thermore we obtain a non–existence theorem for radial ground states using atechnique of Ni and Serrin [13].

1. Introduction. Recently, in [17] for the p–Laplacian equation with weights inRn, under general conditions for the non–linearity f , uniqueness of ground statesand various qualitative properties of solutions were established. Here we proveexistence of crossing radial solutions for f positive near u = 0, and existence ofradial ground states in Rn for f negative near u = 0 of such spatially dependentequations. More specifically, we use a unified proof and a new subcritical condition(Φ) on f at infinity, which was introduced in [1]. Indeed, in canonical cases, (Φ) isinteresting in applications and (Φ) is more general than the well known subcriticalcondition (Φ1) of Castro and Kurepa [4], adopted in several related papers, as [22],[8] and [12].

In particular, we are interested in finding sufficient conditions for existence ofradial ground states of the singular quasilinear elliptic equation with weights

div(g(|x|)|Du|p−2Du) + h(|x|)f(u) = 0 in Rn \ 0,p > 1, n ≥ 1,

(1)

where g, h : R+ → R+ and Du = (∂u/∂x1, · · · , ∂u/∂xn), when f < 0 near u = 0.By a ground state we mean a non–negative non–trivial solution of (1) which tendsto zero at infinity. In the radial form (1) becomes

[a(r)|u′|p−2u′]′ + b(r)f(u) = 0 in R+, p > 1, (2)

2000 Mathematics Subject Classification. Primary: 35J70; Secondary: 35J60.Key words and phrases. Ground states, p–Laplacian operator, weight functions.This research was supported by the Italian MIUR project titled “Metodi Variazionali ed

Equazioni Differenziali non Lineari”.

447

448 E. CALZOLARI, R. FILIPPUCCI AND P. PUCCI

where a(r) = rn−1g(r), b(r) = rn−1h(r), n ≥ 1 and r = |x|. The simple Laplace–Poisson equation arises when a(r) = b(r) = rn−1, where n is the underlying spacedimension.

Moreover, with the same technique, when f > 0 near u = 0, we are also able toprove the existence of a radial crossing solution of (1) in its maximal continuationinterval where u > 0 and u′ < 0. We recall that these existence results wereestablished in [1] for general quasilinear elliptic equations without weights, while in[22] for the p–Laplacian equation without weights.

In addition to the ground state problem, when f < 0 near u = 0, we can alsoconsider existence of non–trivial radial solutions of the homogeneous Dirichlet–Neumann free boundary problem

div(g(|x|)|Du|p−2Du) + h(|x|)f(u) = 0 in BR \ 0,u ≥ 0, u 6≡ 0, u = ∂νu = 0 on ∂BR,

(3)

for some R > 0.A number of examples fall into the general category of (1). A first is the ce-

lebrated Matukuma equation and several generalizations of it in stellar dynamics,cfr. [11], [2], [23], [5], [10], [7] and [14]–[17]. All these models are discussed in detailin Section 4 of [17], as special cases of the main example introduced in [17]

div(rk|Du|p−2Du) + r`

(rs

1 + rs

)σ/s

f(u) = 0,

p > 1, n ≥ 1, k ∈ R, ` ∈ R, s > 0, σ > 0.

(4)

In particular, under the following general conditions on the exponents

` ≥ k − p,k

p+

`

p′≥ 1− n, (5)

where p′ is the Holder conjugate of p, and on f

(f1) f ∈ C(R+) ∩ L1[0, 1],

in [17] a careful definition of semi–classical solution for (1) was given, as well as aqualitative theory. Finally the main uniqueness theorem of [17] can be applied to(4) under appropriate behavior of the non–linearity f(u), satisfied i.e. by

f(u) = −um + u℘; p ≥ 2, −1 < m < ℘ ≤ p− 1, m ≤ 1 +p− 3p− 1

℘. (6)

In other words radial non–negative non–singular semi–classical ground states forequations of type (4)–(6) are unique.

Some existence and non–existence results for radial ground states of special casesof (2) are given in [5] when f is continuous also at u = 0 and non–negative for u > 0small. In the recent paper [9] some existence, non–existence and uniqueness resultsfor radial ground states of some special cases of (2) are given when f > 0 everywherein R+ but singular at u = 0. For a more detailed discussion and comparison withour results we refer to the Remarks after Theorems 5 and 6 in Section 7. Weemphasize, however, that the main case treated in the present paper is when f < 0for u > 0 small, as in [17].

Throughout the paper we shall adopt the definition of semi–classical solutionfor (1) proposed in [17], when f satisfies (f1) and F (u) =

∫ u

0f(v)dv denotes the

p–LAPLACIAN ELLIPTIC EQUATIONS WITH WEIGHTS 449

well defined integral function of f . In the main existence theorems of Section 7 wesuppose also that either

(f2) there exists β > 0 such that F (u) < 0 for 0 0,

as in [8], [12] and [1], or

(f3) there exists c > 0, possibly infinite, such that f(u) > 0 for 0 < u < c,

as in [22] and [1].

Using the main change of variable of [17], we transform (2) into the equivalentequation

[q(t)|vt|p−2vt]t + q(t)f(v) = 0, (7)

that is, into an equation of type (2), but with the same weights. In the specialcase when (7) arises with q(t) = tN−1 for some N ≥ 1, then earlier theory can beapplied, see e.g. [18] and [19], but of course, in general q is no longer a pure power.In order to study the existence of semi–classical solutions of (2), we ask that thetransformed equation (7) is compatible with the basic structure assumptions of [17],namely:

(q1) q ∈ C1(R+), q > 0, qt > 0 in R+;

(q2) qt/q is strictly decreasing on R+;

(q3) limt→0+

tqt(t)q(t)

= N − 1 ≥ 0.

The paper is organized as follows: in Section 2 the definition of semi–classicalsolutions of (1) and preliminary qualitative properties for such solutions are given.In Section 3 we present and summarize the main properties of solutions of thecorresponding initial value problem, in the spirit of [6]. Section 4 is devoted toshowing the connections between the following subcritical growth conditions (Φ1)and (Φ), with Q(t) =

∫ t

0q(τ)dτ , d = β under (f2) and d = 0 under (f3).

(Φ1) The function Φ(v) = pNF (v)− (N −p)vf(v), v ∈ R+, is locally bounded nearv = 0 and there exist µ > d and λ ∈ (0, 1) such that Φ(v) ≥ 0 for all v ≥ µand

lim supv→∞

Φ(λ1v)Q

(Cλ

[vp−1

f(λ2v)

]1/p)

= ∞ for all λ1, λ2 ∈ [λ, 1],

where Cλ = [(1− λ)p′]1/p′ .

Property (Φ1) is equivalent to the famous condition of Castro and Kurepa [4], usedin [22], [8], [12] and [1] in the standard case, when q(t) = tN−1. While property

(Φ) The function Φ(v) = pNF (v)− (N −p)vf(v), v ∈ R+, is locally bounded nearv = 0 and there exist µ > d and λ ∈ (0, 1) such that Φ(v) ≥ 0 for all v ≥ µand

lim supv→∞

Φ(λ1v)Q(cλ[vp+1f(λ2v)]1/p

)= ∞ for every λ1, λ2 ∈ [λ, 1],

where cλ = [(1− λ)p′]1/p′(1− λ)2/p,


is equivalent, when q(t) = tN−1, to the condition introduced in [1]. Of coursecλ < Cλ.

Section 4 ends with some remarks about the independence of the two differentgrowth hypotheses (Φ) and (Φ1). Proposition 1 shows that, under (f1) and the as-sumption lim infu→∞ f(u) = k0 > 0, with k0 possibly ∞, condition (Φ1) is strongerthan (Φ). Examples illustrating the independence of (Φ) and (Φ1) are given inSection 5. For instance, when 1 > 1, with 1 < ℘ < p∗N , where p∗N = Np/(N − p), seeSection 5 and also [1]. In Section 6 some preliminary lemmas are presented to sim-plify the main proofs. In Section 7 existence of crossing solutions is established when(f3) holds, as well as the principal existence theorems if (f2) is verified. Finally,in Section 8 a non–existence theorem for positive radial semi–classical non–singularground states of (1) is given under condition (f2).

From the main results of Sections 7–8, the following consequence can be derived.

Corollary 1. Suppose that 1 < p < N . Assume (q1)–(q3) and let

f(u) = −um + u℘, −1 < m < ℘. (8)

(i) There exists a semi–classical non–singular radial ground state u of (1) when-ever

0 < m < ℘ < p∗N − 1,

and

(q4) limt→∞

q′(t)q(t)

= 0,

holds. Moreover, u is positive in the entire Rn if and only if m ≥ p− 1, while it iscompactly supported when 0 < m < p− 1.

(ii) There exists a positive radial semi–classical non–singular solution of the cor-responding homogeneous Dirichlet–Neumann free boundary value problem (3) if

−1 < m < p− 1, ℘ < p∗N − 1,

and (q4) is valid.(iii) If ℘ ≥ p∗N − 1, then problem (1) admits no positive semi–classical non–

singular radial ground states, when

(q5)Q(t)q′(t)

q2(t)≥ N − 1

Nfor all t > 0

is satisfied.Moreover, in cases (i) and (ii), if for some ν < 1

a(r)b(r)

=g(r)h(r)

∼ c rν as r → 0+, c > 0,

then the constructed solution u of (1) and (8) is regular, that is Du is Holdercontinuous at x = 0, with Du(0) = 0; while if ν ∈ [1, p), then u is Holder continuousat x = 0. In both cases

u ∈ W 1,ploc (Rn),

when also 1 < p ≤ n.

Conditions (q1)–(q5) are given in terms of the main radial weights a and b in thepaper, cfr. (A1)–(A4) in Section 2, (A5) in Section 3 and finally (A6) in Section 8as well as their related comments.


2. Semi-classical solutions. Consider the quasilinear singular elliptic equation

div(g(|x|)|Du|p−2Du) + h(|x|)f(u) = 0

in Ω = x ∈ Rn \ 0 : u(x) > 0; p > 1, n ≥ 1,

u ≥ 0, u 6≡ 0 in Rn \ 0,(9)

where g, h : R+ → R+. Prototypes of (9), with non–trivial functions g, h, are given,for example, by equations of Matukuma type and equations of Batt–Faltenbacher–Horst type, see (1.4) and Section 4 of [17]. In several interesting cases g can besingular at the origin, and in general h also may be singular there; thus it is necessaryin (9) that Ω excludes the point x = 0 and also points where u(x) = 0 because ofthe assumption (f1) which allows f to be singular at u = 0.

We shall be interested in the radial version of (9), namely

[a(r)|u′|p−2u′]′ + b(r)f(u) = 0

in J = r ∈ R+ : u(r) > 0, p > 1, r = |x|,u = u(r), u ≥ 0, u 6≡ 0 in R+,

(10)

where, with obvious notation,

a(r) = rn−1g(r), b(r) = rn−1h(r). (11)

As in [17], in order that the transformed equation (7) should satisfy the requirements(q1)–(q3) we shall ask that the coefficients a, b have the following behavior

(A1) a, b > 0 in R+, a, b ∈ C1(R+),

(A2) (b/a)1/p ∈ L1[0, 1] \ L1[1,∞),

(A3) the function

ψ(r) =[1p· a′

a+

1p′· b′

b

](a

b

)1/p

is positive and strictly decreasing in R+, where p′ is the Holder conjugate of p (> 1),

(A4) there is N ≥ 1 such that

limr→0+

ψ(r)∫ r

0

( b

a

)1/p

= N − 1.

In Section 4 of [17] several equations, such as (4), satisfying the above conditionsand modelling physical phenomena, are presented. As noted in [17], in the specialcase when g ≡ 1 or equivalently a(r) = rn−1, assumptions (A1) and (A2) alsoappear in [10], though in somewhat different circumstances, see also [15] and [16].

As noted above, since (9) is possibly singular when x = 0 and when u = 0, it isnecessary to define carefully the meaning to be assigned to solutions of (9), and inturn, of (10). One can consider weak distribution solutions of (9), or alternativelydistribution solutions with suitable further regularity conditions and well definedvalues at x = 0. Following [17] we shall thus consider the following definition.

Definition. A semi–classical radial solution of (9) is a non–negative function uof class C1(R+), which is a distribution solution of (10) in J , that is for all C1


functions ϕ = ϕ(r), having compact support in J , it results∫

J

rn−1g(r)|u′|p−2u′ ϕ′ dr =∫

J

rn−1h(r)f(u)ϕdr.

In Proposition 2.1 of [17] it is proved that every semi–classical radial solution be-comes a classical solution in R+ when f is continuous in R+

0 with f(0) = 0. Condi-tions which guarantee non–singular behavior of solutions of (10) at r = 0 are alsogiven in [17].

As noted in the Introduction, equation (2) can be transformed in equation (7)by the following change of variables of [17]

t(r) =∫ r

0

[b(s)/a(s)]1/pds, r ≥ 0. (12)

Of course t : R+0 → R+

0 , t(0) = 0, t(∞) = ∞, by (A2), and t is a diffeomorphism ofR+

0 into R+0 by (A1), with inverse r = r(t), t ≥ 0. The relation between the original

weights a and b and the new weight q is given by

q(t) = [a(r(t))]1/p[b(r(t))]1/p′ , t > 0. (13)

Obviously, if u = u(r) is a semi–classical solution of (10) in J , then v = v(t) =u(r(t)) is of class C1(R+) and it satisfies (7) in I = t ∈ R+ : v(t) > 0, namelyv is a semi–classical solution of (7). We emphasize that q ∈ C(R+

0 ) ∩ C1(R+) bycondition (q1). In particular v satisfies (7) in I in the classical sense with v ≥ 0 and

v ∈ C1(R+0 ), |vt|p−2vt ∈ C1(I). (14)

For details see Proposition 3.1 and Theorem 3.2 of [17] together with other relatedresults contained in Section 3 of [17].

In the main example (4) we have

a(r) = rn+k−1, b(r) = rn+`−1

(rs

1 + rs

)σ/s

.

As shown in Section 4 of [17] conditions (A1)–(A4) are satisfied if (5) is verified.Furthermore,

ψ(r) =(

n− 1 +k

p+

`

p′+

σ

p′· 11 + rs

)·(

1 + rs

rs

)σ/ps

· r(k−`)/p−1, (15)

and the limit value N − 1 in (A4) is given by(

n− 1 +k

p+

`

p′+

σ

p′

)· lim

r→0+r(k−`−σ)/p−1

∫ r

0

t(σ−k+`)/pdt, (16)

which immediately yields

N = pn + ` + σ

p + ` + σ − k> 1, (17)

by (5).Finally 1 p − n. The latter condition implies that

a−1/(p−1) 6∈ L1[0, 1]. In this case, as proved in [17] via the main result of [7], thenatural Sobolev exponent of (4) and its transformed equation (7) is given by

1p∗N

=1p− 1

N=

1p· n + k − p

n + ` + σ.

Here p∗N > p because N > 1.


To obtain the asymptotic behavior of u′(r) for r near 0 one can apply Theorem 3.2of [17], since

g(r)h(r)

=a(r)b(r)

∼ rν as r → 0+, ν = k − `− σ,

and in turn (17) can be rewritten in the form

N = pn + k − ν

p− ν.

Thus from (3.10) of [17] we find as r → 0+

r−(1−ν)/(p−1)u′(r) → −[sgn f(α)]( |f(α)|

n + k − ν

)1/(p−1)

. (18)

It is finally interesting in this example that the parameter s in (4) does not appearin any of the exponent relations (5), (17)–(18). This is a reflection of the fact thatthe term rs/(1 + rs) in (4) can be replaced by more general functions having thesame asymptotic behavior.

Conditions (5) have the first consequence that ` ≥ −n. Moreover, either a canbe discontinuous (k < 1 − n) or b discontinuous (−n ≤ ` < 1 − n), but not both inview of the second condition of (5). One can show that necessarily N p when b is discontinuous (ifk > p− n).

For example a is discontinuous if n = 3, p = 2, k = −5/2, ` = −1/2, while b isdiscontinuous and N > p when n = 3, p = 2, k = −1/2, ` = −9/4 and 0 < σ < 1/4.

The well–known Matukuma model is the subcase of (4) when k = 0, −` = σ = s,and the exponent conditions for (A1)–(A4) reduce simply to p ≥ σ and n ≥ 1+σ/p,with N = n. Here ν = k− `− σ = 0, we get u′(r) = O(r1/(p−1)) and so u(r)− α =O(rp/(p−1)). In particular u ∈ C1(Rn).

For the standard Matukuma equation, namely when k = 0, p = 2, −` = σ = s =2 and n = 3, the transformed equation (7) arises with q(t) = sinh2 t/ cosh t. In thiscase we have N = n = 3, and so the critical Sobolev exponent is 2∗3 = 6, the usualcritical exponent for the Matukuma equation in R3, as well known in the literature.

The case σ = 0 in (4) is not allowed. If, nevertheless, we do set σ = 0 thenconditions (A1)–(A4) will be satisfied if the relations of (5) hold as strict inequalities,with

N = pn + `

p + `− k> 1.

This is clear in the main example of [18], see also [17], when in (9)

g(r) ≡ 1, h(r) = r`.

Indeed, here σ = 0 and (A1)–(A4) hold if

` + p > 0, ` + (n− 1)p′ > 0, (19)

with

N = p` + n

` + p> 1. (20)

At the same time using the main change of variable (12) we see that (7) takes thecanonical form

[tN−1|vt|p−2vt]t + tN−1f(v) = 0,

by (12) and (13), that is, N serves as the natural dimension for this example.


For this case we have n > p if and only if N > p. This condition also impliesthat a−1/(p−1) = r−(n−1)/(p−1) 6∈ L1[0, 1]. Thus the natural Sobolev exponent is

1p∗N

=1p− 1

N=

n− p

` + n,

confirming again the role of N as the natural dimension of the problem.For other equations modelled by (4) we refer to Section 4 of [17].

3. Preliminary Results. For simplicity, from this point on we write ′ = d/dt ifthere is no confusion in the notation. In this section, as in [1], we present somepreliminary results useful for the proof of the main existence theorems of crossingsolutions and of radial ground states of (2) via equation (7). In particular, they aresemi–classical solutions of the initial value problem

[q(t)|v′(t)|p−2v′(t)]′ + q(t)f(v) = 0, t > 0,

v(0) = α, v′(0) = 0.(21)

Define

d :=

β, if (f2) holds,0, if (f3) holds.

and γ := supv > d : f(u) > 0 for u ∈ (d, v).

Moreover, as in [1], from now on we assume together with (f1) also(f4) f ∈ Liploc(0, γ).

Finally, we restrict our attention to solutions v of (21), with

α ∈ (d, γ). (22)

Lemma 1. Assume that f satisfies either (f2) or (f3). If v is a semi–classicalsolution of (21) and (22), then v′(t) < 0 near the origin. Moreover v is uniqueuntil it exists and remains in (0, γ), provided that v′(t) < 0.

Proof. From (22), we deduce that there exists t0 > 0 sufficiently small such that

[q(t)|v′(t)|p−2v′(t)]′ = −q(t)f(v) < 0, t ∈ (0, t0).

Hence q(t)|v′(t)|p−2v′(t) is strictly decreasing in (0, t0) and it assumes value zero att = 0 from (q1). Consequently v′(t) < 0, t ∈ (0, t0). Thus v is a solution of the firstorder differential system

w′(t) = −q′(t)q(t)

w(t)− f(v)

v′(t) = −[−w(t)]1/(p−1)

v(0) = α, w(0) = 0,

(23)

where we have used the fact that sgn w(t) = sgn v′(t), being w(t) = |v′(t)|p−2v′(t).Finally, (q1) and (f4) guarantee that (23) admits a unique solution such that

v(t) > 0 and −∞ < v′(t) < 0

for all t > 0.

Lemma 1 says that the unique local solution vα of (21) and (22) can be continuedexactly until tα ≤ ∞, where tα is the first point in R+, uniquely determined, suchthat either vα(tα) = 0 and v′α(tα) ≤ 0, or vα(tα) > 0 and v′α(tα) = 0.


Let Iα = (0, tα) denote the maximal interval of continuation of every semi–classical solution of (21)–(22), under the restrictions

vα > 0 and −∞ < v′α < 0 in Iα. (24)

From the definition of Iα, it is clear that the solutions of (21) we consider have theproperty that v′(t) < 0. Letting ρ(t) = |v′(t)|, problem (21)–(22) can be rewrittenas

[q(t)ρ(t)p−1]′ = q(t)f(v), t ∈ Iα,

v(0) = α ∈ (0, γ), v′(0) = 0.(25)

In analogy to [1] and [12] we give the following lemmas.

Lemma 2. Let v1 be a solution of (25) defined in its maximal interval I1,α, deter-mined by (24). For all t0 ∈ I1,α and ε > 0, there exists δ > 0 such that, if v2 is asolution of (25), with |v1(0)− v2(0)| < δ, then v2 is defined in [0, t0] and

sup[0,t0]

|v1(t)− v2(t)|+ |v′1(t)− v′2(t)| < ε.

Proof. By using (q1)–(q3), the proof of Lemma 2.3 of [12] for the special caseq(t) = tN−1 can be repeated since it was used only the fact that f ∈ Liploc(0, γ)together with (f1). For a complete proof we refer to Lemma 4.2 of [3].

A natural energy function associated to solutions v of (7) is given by

E(t) =ρp(t)

p′+ F (v(t)), ρ = |v′|, (26)

which is of class C1(I ∪ 0), with E′(0) = 0 and in I

E′(t) = −q′(t)q(t)

ρp(t), E(t)− E(s0) = −∫ t

s0

q′(s)q(s)

ρp(s)ds, (27)

see Lemma 5.3 and Section 5 of [17] for more detailed properties.

Lemma 3. Suppose that f satisfies alternatively either (f2) or (f3). Let v be asolution of (25). Then the following results hold.

(i) The limit`α := lim

t→t−αv(t) (28)

belongs to [0, β) if (f2) holds; while `α = 0 if (f3) holds.(ii) If `α > 0 and tα ≤ ∞, then lim

t→t−αv′(t) = 0.

(iii) If tα = ∞, then limt→∞

v′(t) = 0.

(iv) Let λ > d. If α > λ, then there exists a unique value tλ,α ∈ Iα such thatv(tλ,α) = λ.

Proof. (i) Clearly the limit in (28) exists and is non–negative, since v is strictlydecreasing and positive in Iα by (24). Suppose first that (f2) holds. Then `α ∈ [0, γ)by (22). Assume by contradiction that `α ∈ [β, γ). Then β ≤ `α < v(t) < α in Iα,and in turn, by (25) and (f2), from [q(t)|v′(t)|p−1]′ = q(t)f(v) > 0, it follows thatq|v′|p−1 is strictly increasing in Iα.

Distinguish now two cases: tα < ∞ and tα = ∞. If tα < ∞, since v(tα) =`α ≥ β > 0, then v′(tα) = 0 by (24), and so q(t)|v′(t)|p−1 → 0 as t → t−α . On theother hand (q|v′|p−1)(0) = 0, and this contradicts the fact that q|v′|p−1 is strictlyincreasing in Iα.


While if tα = ∞, then Iα = R+ and by Lemma 5.3 of [17] the energy functionalong v defined in (26) is of class C1(R+), with

E′(t) = −q′(t)q(t)

ρp(t) < 0 in Iα. (29)

Therefore E admits finite limit as t → ∞. By (26) also ρ(t) has limit as t → ∞,and

v′(t) −→t→∞

0, (30)

since `α ∈ [β, γ) by contradiction. Rewrite the equation in (25) in the equivalentform

[ρp−1(t)]′ +q′(t)q(t)

ρp−1(t) = f(v(t)), t ∈ Iα. (31)

By (30) and (q2)q′

q[ρ(t)]p−1 −→

t→∞0,

and in turn by (31)lim

t→∞[ρp−1(t)]′ = f(`α) > 0,

since `α ∈ [β, γ) by contradiction. This is impossible, since ρp−1 > 0 in R+ andρp−1(t) → 0 as t →∞.

Suppose now that (f3) holds and that `α > 0 by contradiction. We can repeatthe above proof, with [β, γ) replaced by (0, γ), and obtain the desired contradiction.

(ii) If `α > 0 and tα < ∞, then v′(t) → v′(tα) = 0 as t → t−α by (24). While if`α > 0 and tα = ∞, arguing as in (i), case tα = ∞, and using (29), we obtain thevalidity of (30).

(iii) If tα = ∞, by (i) and the fact that E admits limit as t →∞ then v′(t) → 0as t →∞.

(iv) In this case the proof is an immediate consequence of the fact that v isstrictly decreasing in Iα by (24).

As in [1] define

I− := α ∈ (d, γ) : tα < ∞, `α = 0, v′α(tα) < 0. (32)

Lemma 4. Suppose that f verifies either (f2) or (f3). Let v be a semi–classicalsolution of (25), in Iα defined in (24). If α /∈ I−, then for all t ∈ Iα

q(t)q′(t)

<v(t)

F (v(t))p′[F + F (v(t))]

1/p′,

where F := max[0,d]

F−(v), with F− := max−F, 0.

When f satisfies (f3), clearly F = 0.

Proof. Let α /∈ I− and suppose by contradiction that there exists t ∈ Iα such that

q(t)q′(t)

≥ V

F (V )p′[F + F (V )]

1/p′, (33)

where V := v(t). PutM := sup

[t,tα)

ρ(t).

We claim that there is T1 ∈ [t, tα), with M = ρ(T1). Indeed, since α /∈ I− one ofthe following three cases occurs: (tα = ∞) ∨ (`α > 0) ∨ (v′(tα) = 0). In the third


case there is nothing to prove, that is M is achieved in [t, tα); if tα = ∞, then (30)holds by Lemma 3 (iii), that is M is again achieved in [t, tα). Finally if `α > 0,then v′(tα) = 0 by Lemma 3, and in turn M is again achieved in [t, tα).

Moreover

E(tα) =ρp(tα)

p′+ F (`α) = F (`α) ≤ 0. (34)

Indeed if (f2) holds, then `α ∈ [0, β) and E(tα) = F (`α) ≤ 0; while if (f3) holds,E(tα) = F (`α) = 0. Using (q2) and (24)

∫ tα

t

q′(s)q(s)

ρp(s)ds ≤ q′(t)q(t)

Mp−1

∫ tα

t

[−(v′(s))]ds ≤ q′(t)q(t)

Mp−1V. (35)

Hence by (27) and (34)

F (V ) = F (v(t)) < E(t) = E(tα) +∫ tα

t

q′(s)q(s)

ρp(s)ds ≤ q′(t)q(t)

Mp−1V. (36)

Now by (34) and (35)

Mp

p′=

ρp(T1)p′

= E(T1)− F (v(T1)) ≤ E(tα) +∫ tα

T1

q′(s)q(s)

ρp(s)ds + F

≤ F +q′(T1)q(T1)

Mp−1V ≤ F +q′(t)q(t)

Mp−1V.

(37)

By the assumption of contradiction (33)

q(t)q′(t)

≥ V

F (V )· p′[F + F (V )]p′[F + F (V )]

1/p,

and so

p′[F + F (V )] ≥(

p′V

F (V )[F + F (V )]

q′(t)q(t)

)p

= p′V

F (V )[F + F (V )]

q′(t)q(t)

(p′

V

F (V )[F + F (V )]

q′(t)q(t)

)p−1

,

that isF (V )

V· q(t)q′(t)

≥(

p′V

F (V )[F + F (V )]

q′(t)q(t)

)p−1

.

Therefore by (36) we get

Mp−1 >F (V )

V· q(t)q′(t)

≥(

p′V

F (V )[F + F (V )]

q′(t)q(t)

)p−1

,

in other wordsM

p′>

V

F (V )[F + F (V )]

q′(t)q(t)

. (38)

Consequently by (37)

F ≥ Mp

p′− q′(t)

q(t)Mp−1V = Mp−1

[M

p′− q′(t)

q(t)V

],


and in turn by (36) and (38)

F > Mp−1

[V

F (V )(F + F (V ))

q′(t)q(t)

− Vq′(t)q(t)

]= Mp−1V

q′(t)q(t)

[F + F (V )

F (V )− 1

]

≥ F (V )V

· q(t)q′(t)

· V q′(t)q(t)

[F + F (V )− F (V )

F (V )

]= F.

This is the desired contradiction.

LetΦ(v) := pNF (v)− (N − p)vf(v), v ∈ R+, (39)

be the function given in assumptions (Φ1) and (Φ) of the Introduction, and alonga solution v of (25) in Iα let P be defined as

P (t) := (N − p)q(t)v(t)v′(t)|v′(t)|p−2 + pQ(t)E(t), Q(t) =∫ t

0

q(s)ds. (40)

Clearly P (0) = 0 by (f1), since v′(0) = 0 and Q(0) = 0.We present an inequality proved with the same technique of Lemma 2.4 of [12].

Lemma 5. (Ni–Pucci–Serrin) Assume 1 < p < N . Let v be a solution of (25) inIα, given in (24). Then

P (t) ≥∫ t

0

q(τ)[Φ(v(τ))− p2F (v(τ))]dτ. (41)

Proof. By Lemma 5.3 of [17] the energy function E is of class C1(Iα) and by theregularity of v in Iα we can differentiate P in (40), obtaining in Iα by (29)

P ′(t) = (N − p)

q|v′|p + v[qv′|v′|p−2

]′+ pqE − p

Qq′

q|v′|p

= (N − p)q

pF (v)− vf(v) + p

(1− Qq′

q2

)|v′|p

,

where we have used (25) and (26) for the last step. Adding and subtracting theterm (p∗N−p)F (v), where p∗N = pN/(N−p) represents the Sobolev critical exponentfor (25), as discussed and proved in Section 4 of [17], we get

P ′(t) = (N − p)q

pN

N − pF (v)− vf(v)− (p∗N − p)F (v) + p|v′|p

(1− Qq′

q2

).

By (39) and the fact that Qq′/q2 ≤ 1 in R+ by (q1)–(q3), we have

P ′(t) ≥ qΦ(v)− (p∗N − p)(N − p)qF (v) = q[Φ(v)− p2F (v)].

Finally (41) follows at once since P (0) = 0, as noted above.

We recall that throughout the section we continue to assume the validity of (f1)and (f4).

Theorem 1. Let 1 < p < N and γ = ∞. Suppose that f satisfies (f3), (Φ) and

(f5) lim infv→0+

vp

F (v)= 0.

Then I− 6= ∅.


Proof. Suppose by contradiction that I− = ∅. Hence

α /∈ I− for all α ∈ R+. (42)

Take µ and λ as required in (Φ), and α > 0 so that α > µ > d = 0 by (f3). Withoutloss of generality we suppose that λ is so close to 1 and µ > 0 so close to 0 that

α >µ

λ, 0 <

α(1− λ)tλ

=v(0)− v(tλ)

tλand

µp′1/p′

[F (µ)]1/p∈ Ψ(R+), (43)

with Ψ = q/q′ and where tλ, tµ ∈ Iα are by Lemma 3 (iv) the unique points t suchthat v(tλ) = λα and v(tµ) = µ. Indeed, (43)3 holds for µ > 0 sufficiently smallby (f5) and the fact that Ψ(0) := limt→0+ Ψ(t) = 0 by (q3) since N > 1. Clearlytλ < tµ < tα, since v(tλ) = λα > µ = v(tµ), by (43) and the fact that v′ < 0 in Iα,and

v(0)− v(tλ)tλ

≤ 1,

since

0 = v′(0) = limt→0+

v(t)− v(0)t

= limλ→1

v(tλ)− v(0)tλ

.

Indeed if λ → 1, that is if λα → α, then tλ → 0.Integrating the equation in (25) on [0, t], with t ∈ (0, tλ), we obtain

q(t) · [ρ(t)]p−1 =∫ t

0

q(τ)f(v(τ))dτ, (44)

since ρ(0) = |v′(0)| = 0. Hence, putting

f(λ2α) := max[λα,α]

f(u), λ2 ∈ [λ, 1],

we have f(λ2α) > 0, since λα > µ > 0. Therefore by (44)

q(t) · [ρ(t)]p−1 ≤ max[0,tλ]

f(v(t))∫ t

0

q(τ) dτ ≤ f(λ2α)Q(t),

and in turn

|v′(t)| ≤[f(λ2α) · Q(t)

q(t)

]1/(p−1)

≤ [f(λ2α) · t]1/(p−1),

since Q(t)/q(t) ≤ t by (q1). Integrating this inequality on [0, tλ],∫ tλ

0

−v′(s)ds ≤∫ tλ

0

[s · f(λ2α)]1/(p−1)ds

by (24). Thus

−v(tλ) + v(0) = α(1− λ) ≤ [f(λ2α)]1/(p−1) · tp′

λ

p′,

in other words[f(λ2α)]1/(p−1) ≥ α(1− λ)p′ · t−p′

λ . (45)We choose λ so close to 1 that α(1− λ)f(λ2α) ≤ 1. Therefore

[α(1− λ)f(λ2α)]1/(p−1) ≤ 1

[α(1− λ)f(λ2α)]1/(p−1)≤

[1

α(1− λ)

]1/(p−1)tp′

λ

α(1− λ)p′,

by (45). Consequently

tλ ≥ (1− λ)1+1/p(p′)1/p′ [αp+1f(λ2α)]1/p

= cλ

[αp+1f(λ2α)

]1/p, (46)


where cλ is the constant given in (Φ).Since α /∈ I− and F = 0 by (f3), from Lemma 4

q(tµ)q′(tµ)

<µ[p′F (µ)]1/p′

F (µ)=

µp′1/p′

[F (µ)]1/p. (47)

Therefore by (q3) and the fact that q/q′ is invertible by (q2), we have by (43)

tµ < Ψ−1

(µp′1/p′

[F (µ)]1/p

):= cµ. (48)

ClearlyΦµ := inf

0<v≤µΦ(v) > −∞,

since Φ is locally bounded near 0 and Φ ∈ C(R+) by (Φ). Furthermore, sinceα > λα > µ by (43) and (Φ), there is λ1 ∈ [λ, 1] such that

Φ(λ1α) := minλα≤v≤α

Φ(v) ≥ 0.

By construction we now have

Φ(v(t)) ≥

Φ(λ1α), if 0 < t < tλ,0, if tλ ≤ t ≤ tµ,−|Φµ|, if t > tµ.

(49)

Since 1 < p < N , by (40), (24), Lemma 5, (49) and (q1) for all t ≥ tµ

p(N − p)Q(t)E(t) ≥ P (t)

≥(∫ tλ

0

+∫ tµ

tλ

+∫ t

tµ

)q(s)Φ(v(s))ds− p2

∫ t

0

q(s)F (v(s))ds

≥ Φ(λ1α)Q(tλ)− |Φµ|Q(t)− p2F (µ)Q(t),

since F is strictly increasing in R+0 by (f3) and the assumption γ = ∞, so that

F (v(s)) ≤ F (µ) for all t ∈ (tµ, tα). (50)

Hence by (46) for all t ∈ (tµ, tα)

p(N − p)E(t) ≥ Φ(λ1α)Q(cλ[αp+1f(λ2α)]1/p)

Q(t)− |Φµ| − p2F (µ). (51)

We now treat the cases tα < ∞ and tα = ∞ separately.Assume first that tα < ∞. For each ε > 0 define

Tε := mintµ + ε, tα,so that

Tε ∈ (tµ, tα]. (52)By (Φ) we can take α > µ/λ so large that by (51)

E(t) ≥ F (µ) +1p′·(µ

ε

)p

in (tµ, Tε]. (53)

In particular for t = Tε

E(Tε) ≥ F (µ) +1p′·(µ

ε

)p

> F (µ). (54)

We claim thatTε = tµ + ε < tα. (55)


Assume for contradiction that Tε = tα < ∞, then v′(Tε) = v′(tα) = 0. Indeed`α = 0 in (28) by (f3) and Lemma 3 (i). Furthermore, since α /∈ I−, then v′(tα) = 0.Hence by (26) and (50)

E(Tε) = F (v(Tε)) ≤ F (µ),which contradicts (54) and proves the claim (55).

By (53) for all t ∈ (tµ, Tε] ⊂ Iα

|v′(t)|pp′

+ F (v(t)) ≥ F (µ) +1p′

(µ

ε

)p

,

and −v′(t) = |v′(t)| > µ/ε by (50). Integrating in (tµ, Tε] by (55)

v(tµ)− v(Tε) >µ

ε(Tε − tµ) = µ,

that is v(Tε) < 0. This is impossible since v > 0 in [0, tα) and completes the proofin the case tα < ∞.

Assume next that tα = ∞. By (51), (26), (50) and the assumption 1 < p < N ,for all t ≥ tµ

(N − p)(p− 1)|v′|p ≥ Φ(λ1α)Q(tλ)Q(t)

− |Φµ| − p2F (µ)− p(N − p)F (µ). (56)

Since (tµ, tµ + 1) ⊂ Iα, by (56) for all t ∈ (tµ, tµ + 1)

(N − p)(p− 1)|v′|p ≥ Φ(λ1α)Q(tλ)

Q(cµ + 1)− |Φµ| − pNF (µ),

where cµ is the number defined in (48), which depends only on µ, p, F , but isindependent of α, and we have used (q1) to have that

Q(tµ) ≤ Q(t) ≤ Q(tµ + 1) < Q(cµ + 1) in (tµ, tµ + 1),

by (42) and by Lemma 4. By (Φ) we can take α so large that |v′| ≥ µ in [tµ, tµ +1].By (24) and integration on [tµ, tµ + 1], we get

v(tµ + 1) = v(tµ) +∫ tµ+1

tµ

v′(s)ds = v(tµ)−∫ tµ+1

tµ

|v′(s)|ds ≤ µ− µ = 0,

which contradicts the fact that v > 0 in Iα = R+ and completes the proof also inthe case tα = ∞.

Remark. If f(v) ∼ vm as v → 0+, with m > −1 by (f1), then (f5) holds if andonly if m < p− 1.

The assertion of Theorem 1 continues to hold also when γ < ∞, and under thevalidity of (f2), when condition (f5) is replaced by (q4), as shown in the next

Theorem 2. Assume that f satisfies (f1), (f4) and either (f2) or (f3). Supposealso that q verifies condition (q4). If (Φ) holds, then I− 6= ∅.Proof. Case 1: γ < ∞. Problem (25) admits a solution v by Proposition 1, whichis unique until it exists and its range remains in the interval where f is locallyLipschitzian, that is in (0, γ), by (f4). By Lemma 2, with

v1(t) = v(t, α) and v2(t) ≡ γ,

andlimα→γ

v(t, α) = γ,


uniformly in any bounded interval of R+0 . Furthermore taking α sufficiently close

to γ so that α > γ, where γ := (γ + d)/2, by (iv) of Lemma 3 there is a uniquevalue tα ∈ Iα such that

v(tα, α) = γ.

We claim that the function defined in (d, γ) by

α 7→ tα

is not bounded as α → γ. Indeed, if there is a constant t, such that 0 < tα ≤ t < ∞,as α → γ, all the corresponding solutions v(t, α) would be bounded above by γ byt ≥ t, since v is decreasing in Iα. This contradicts the fact that v(t, α) converges toγ as α → γ, uniformly on any bounded interval of R+

0 .Hence there exists α ∈ (d, γ) such that the corresponding solution v(t, α), with

v(tα, α) = γ, satisfies the following property

tα > Ψ−1

(γ

F (γ)p′[F + F (γ)]1/p′

),

by (q4). Lemma 4 guarantees that α ∈ I−, concluding the proof when γ < ∞.Case 2: γ = ∞. We can proceed as in the proof of Theorem 1 until (47), namely

q(tµ)q′(tµ)

<

(µ

F (µ)p′[F + F (µ)]1/p′

)

and now by (q4)

tµ < Ψ−1

(µ

F (µ)p′[F + F (µ)]1/p′

).

At this step we can proceed exactly as in the proof of Theorem 1 with the singleexception that R+ is replaced by (d,∞).

Remarks. Theorem 1 can be applied to the Matukuma equation, but not Theo-rem 2, since

q(t) =sinh2t

cosh tand lim

t→∞q′(t)q(t)

= 1.

However (q4) holds in several cases. For instance, for the general equation (4), sinceq′(t)/q(t) = ψ(r(t)), under conditions (5), by (15) we have

limt→∞

q′(t)q(t)

=

0, if either ` = k − p and k/p + `/p′ = 1− n, orif ` > k − p and k/p + `/p′ ≥ 1− n,

`∞ ∈ R+, if ` = k − p and k/p + `/p′ > 1− n.

In particular in all the cases of (4) in which the parameters verify either

` = k − p andk

p+

`

p′= 1− n, (57)

or` > k − p and

k

p+

`

p′≥ 1− n, (58)

condition (q4) is satisfied.For instance (57) holds when n = p = σ = −` = 3, s = 2 and k = 0 so that

q(t) = tanh2 t; while condition (58) is valid when n = p = 2, σ = s = −` = 1 andk = 0 with q(t) = (t2 + 4t)/2(t + 2).

Assumption (q4) in terms of the radial weights a and b of the original radialequation (2), becomes


(A5) limr→∞

ψ(r) = 0,

since Q(t(r)) =∫ r

0b(s)ds by (40), (13) and (12). In particular in the interesting

subcase of (1) given when g ≡ 1, assumption (A5) reduces to

(A5)′ limr→∞

ψh(r) = 0,where

ψh(r) =[n− 1

r+

1p′

h′

h

]

see also [17].

We now establish the results contained in Theorems 1 and 2 when (Φ) is replacedby condition (Φ1) of Section 1, which was introduced in [4] for the Laplacian equa-tion in a ball. Condition (Φ1) is the analogue subcritical assumption used in [8] and[22] for the p–Laplacian equations with no weights. The results of [8] were extendedin [12] to A–equations, while those of [8] and [22] were extended to A–equationsin [1], with a unified proof and also with the introduction of the new subcriticalcondition (Φ).

Theorem 3. Let γ = ∞. Suppose that f verifies (f1) and (f3)–(f5). If (Φ1)holds, then I− 6= ∅.Proof. We proceed as in the proof of Theorem 3 until (45). Now

tλ ≥ [(1− λ)p′]1/p′ ·[

αp−1

f(λ2α)

]1/p

= Cλ

[αp−1

f(λ2α)

]1/p

,

which replaces (46) in the proof of Theorem 3. Proceeding as in the proof ofTheorem 3 until (51), which becomes

p(N − p)E(t) ≥ Φ(λ1α)Q(t)

Q

(Cλ

[αp−1

f(λ2α)

]1/p)− |Φµ| − p2F (µ).

Also in this case we distinguish the two cases tα < ∞ and tα = ∞, and get thesame conclusions as before.

Theorem 4. Let (f1), (f4), (Φ1) and (q4) hold. If f verifies either (f2) or (f3),then I− 6= ∅.Proof. In analogy of the proof of Theorem 3, following the proof of Theorem 2, wearrive to the desired conclusion.

4. Relation between (Φ) and (Φ1) when γ = ∞. In this section we compare thetwo growth conditions (Φ) and (Φ1), as done in [1] when the weight q is the standardweight rn−1. In particular we shall show that in cases interesting in applications,condition (Φ) holds while (Φ1) does not.

Proposition 1. Let γ = ∞. If

lim infv→∞

f(v) = k0 ∈ (0,∞], (59)

then (Φ1) implies (Φ).


Proof. Fix λ ∈ (0, 1) and first note that for all λ2 ∈ [λ, 1] and v > d/λ

cλ[vp+1f(λ2v)]1/p = Cλ

[vp−1

f(λ2v)

]1/p

· [(1− λ)vf(λ2v)]2/p (60)

By (59) we have lim infv→∞ vf(v) ≥ limv→∞ v · lim infv→∞ f(v) = ∞, so that

limv→∞

vf(v) = ∞.

Hence there is v0 ≥ µ/λ sufficiently large, where µ > d is the number given in (Φ1),such that for all v ≥ v0

f(v) > 0 and vf(λ2v) ≥ λ2vf(λ2v) ≥ 11− λ

. (61)

Thus for v ≥ v0 by (60) and (61)

cλ[vp+1f(λ2v)]1/p ≥ Cλ

[vp−1

f(λ2v)

]1/p

,

that is

Q(cλ[vp+1f(λ2v)]1/p

)≥ Q

(Cλ

[vp−1

f(λ2v)

]1/p)

,

since Q is increasing by (q1). In turn by (Φ1) we have

Φ(λ1v)Q(cλ[vp+1f(λ2v)]1/p

)≥ Φ(λ1v)Q

(Cλ

[vp−1

f(λ2v)

]1/p)

,

and the conclusion follows.

The two growth conditions (Φ1) and (Φ) are independent, since also the reverseimplication of Proposition 1 holds, as shown in the next

Proposition 2. Let γ = ∞. If

lim supv→∞

vf(v) = k1 ∈ [0,∞), (62)

then (Φ) implies (Φ1).

Proof. As in the proof of Proposition 1, fix λ ∈ (0, 1). By (60) for all λ2 ∈ [λ, 1]and v > d/λ we have

Cλ

[vp−1

f(λ2v)

]1/p

=cλ[vp+1f(λ2v)]1/p

[(1− λ)vf(λ2v)]2/p. (63)

By (62) there is ε > 0 such that k1 + ε ≤ λ/(1−λ), and in turn by (62) again thereis v0 ≥ µ/λ sufficiently large, where now µ > d is the number given in (Φ), suchthat for all λ2 ∈ [λ, 1] and v ≥ v0

vf(λ2v) ≤ k1 + ε

λ2≤ 1

1− λ.

Hence by (63)

Cλ

[vp−1

f(λ2v)

]1/p

≥ cλ[vp+1f(λ2v)]1/p.

Therefore, since now for all v ≥ v0 and λ1 ∈ [λ, 1]

Φ(λ1v)Q

(Cλ

[vp−1

f(λ2v)

]1/p)≥ Φ(λ1v)Q

(cλ[vp+1f(λ2v)]1/p

),


the conclusion follows at once, as above.

Remark. When assumption (62) holds with the limsup replaced by the limit, thenlimv→∞ f(v) = 0, and the existence problem could be solved with much simplertechniques. Moreover, the non–linearities which frequently appear in applicationstend to infinity at infinity, a subcase of (59).

Hence in cases interesting in applications (Φ) is more general than (Φ1). Inparticular under (59), Theorems 3 and 4 are immediate corollaries of Theorems 1and 2, since in general (59) together with (Φ), (f1), (f4) and either (f2) or (f3)do not imply the validity of (62). This will be clarified in the next section.

5. Canonical non–linearities in the case γ = ∞. Since in conditions (Φ) and(Φ1) only the behavior of f at infinity is important, in the examples we present inthis section, we shall define the various non–linearities only for large values of v.We also recall that in the sequel 1 < p < N and γ = ∞. To simplify the notationin (Φ) and (Φ1) we shall denote by χ and χ1 the main involved functions, namely

χ(v) := Φ(λ1v)Q(cλ[vp+1f(λ2v)]1/p), χ1(v) := Φ(λ1v)Q

(Cλ

[vp−1

f(λ2v)

]1/p)

.

First consider

f(v) = vp∗N−1 +1v

for v ≥ v0 > 0.

Let f be defined in [0, v0] so that f ∈ L1[0, v0] ∩ C(R+) ∩ Liploc(R+), and also insuch a way that f satisfies (f1), (f4) and either (f2) or (f3) in its entire domainR+, and finally so that Φ is locally bounded near v = 0. Clearly also (59) holdswith k0 = ∞, since 1 < p < N . Moreover for all v ≥ v0

F (v) = F (v0)− vp∗N0

p∗N− log v0 +

vp∗N

p∗N+ log v.

Hence, setting c0 :=[F (v0)− log v0 − v

p∗N0 /p∗N

]Np−N + p, we have for all v ≥ v0

Φ(v) = NpF (v)− (N − p)vf(v) = c0 + pN log v.

Of course Φ is positive for all v sufficiently large, say for v ≥ µ, with µ > maxd, v0.Fix λ ∈ (0, 1). Then for all λ1, λ2 ∈ [λ, 1] and v ≥ µ/λ

χ(v) = [c0 + pN log(λ1v)] ·Q(

cλ

[λ

p∗N−12 vp+p∗N +

vp

λ2

]1/p)→∞

as v →∞, since Q(t) →∞ as t →∞ by (q1), and in turn (Φ) holds.While we claim that (Φ1) does not hold. Indeed,

limv→∞

Cλ

[vp−1

f(λ2v)

]1/p

= limv→∞

Cλλ(1−p∗N )/p2 v1−p∗N /p = 0, (64)

since 1 < p < p∗N . Hence for all v ≥ µ/λ sufficiently large and for all λ2 ∈ [λ, 1] wehave that Cλ

[vp−1/f(λ2v)

]1/p ∈ (0, 1), and so also for all λ1 ∈ [λ, 1]

χ1(v) ≤ q(1)Cλ · [c0 + pN log(λ1v)] · v(p−1)/p

[(λ2v)p∗N−1 +

1λ2v

]1/p,


by (q1). Therefore

0 ≤ limv→∞

χ1(v) ≤ q(1)CλpNλ(1−p∗N )/p2 lim

v→∞v1−p∗N /p log(λ1v) = 0,

and the claim is proved.

Remark. Since the number N , given in (q3), is strictly greater than 1, we claimthat for all ε ∈ (0, N − 1) there is t0 = t0(ε) > 0 and two constants C1,ε, C2,ε > 0,depending on ε and N , such that

C1,εtN+ε ≤ Q(t) ≤ C2,εt

N−ε for t ∈ (0, t0). (65)

Indeed, fixed ε ∈ (0, N − 1), by (q3) there is t0 = t0(ε) > 0 such that for allt ∈ (0, t0)

0 <N − 1− ε

t<

q′(t)q(t)

<N − 1 + ε

t.

Integrating on [s, t0], with s ∈ (0, t0), we get by (q1)(

t0s

)N−1−ε

≤ q(t0)q(s)

≤(

t0s

)N−1+ε

,

that is, putting κ1,ε := q(t0)t1−N−ε0 and κ2,ε := q(t0)t1+ε−N

0 ,

κ1,εsN−1+ε ≤ q(s) ≤ κ2,εs

N−1−ε.

Now, integrating the above inequality from 0 to t, we obtain (65), with

C1,ε :=κ1,ε

N + ε> 0 and C2,ε :=

κ2,ε

N − ε> 0.

Let ℘ ∈ (1, p∗N ),

f(v) = vp∗N−1 + v℘−1 for all v ≥ v0 > 0,

and let again f be defined in [0, v0] so that f ∈ L1[0, v0] ∩ C(R+) ∩ Liploc(R+),and also in such a way that f satisfies (f1), (f4) and either (f2) or (f3) in itsentire domain R+, and finally so that Φ is locally bounded near v = 0. Also (59) issatisfied with k0 = ∞. By (f1) for all v ≥ v0

F (v) = F (v0) +vp∗N

p∗N+

v℘

℘− v

p∗N0

p∗N− v℘

0

℘.

Hence Φ(v) = c0 + c1v℘ for all v ≥ v0, where

c0 := Np

[F (v0)− v

p∗N0

p∗N− v℘

0

℘

]and c1 := p−N +

Np

℘> 0,

since ℘ < p∗N . Thus there is µ > maxd, v0 sufficiently large such that Φ(v) ≥ 0for all v ≥ µ. Fix λ ∈ (0, 1) and put v0 = µ/λ. Therefore for all v ≥ µ/λ and λ1,λ2 ∈ [λ, 1] we have

Φ(λ1v) ·Q(cλ[λp∗N−12 vp+p∗N + λ℘−1

2 vp+℘]1/p) →∞,

as v →∞, namely (Φ) holds.Now, since ℘ < p∗N and p < p∗N ,

limv→∞

Cλ

[vp−1

f(λ2v)

]1/p

= Cλλ(1−p∗N )/p2 lim

v→∞v1−p∗N /p = 0.


Furthermore, since ℘ < p∗N , fix ε ∈ (0, N − 1) so small that

℘ < p∗N − εp

N − p.

Since 1 < p < N , by (65)2 there is v1 = v1(ε) ≥ µ/λ sufficiently large such that forall v ≥ v1 and λ1, λ2 ∈ [λ, 1] we have

Φ(λ1v)·Q(

Cλv(p−1)/p

[(λ2v)p∗N−1 + (λ2v)℘−1]1/p

)≤ C2,εCλ

N−ε [c0 + (λ1v)℘c1] · v(p−1)(N−ε)/p

[(λ2v)p∗N−1 + (λ2v)℘−1](N−ε)/p.

Thus the right hand side approaches zero as v →∞ since

℘ +(p− 1)(N − ε)

p− (p∗N − 1)(N − ε)

p< 0

by the choice of ε. Therefore (Φ1) does not hold.

By Proposition 2 when (62) holds then Theorems 1 and 2 are corollaries ofTheorems 3 and 4, respectively, since (Φ) implies (Φ1). To show this we shallpresent some examples of non–linearities satisfying (62) and (Φ1), but not (Φ).

Let m ≤ −p

f(v) = vm−1, v ≥ v0 > 0, (66)

and let f be defined in [0, v0] so that f ∈ L1[0, v0] ∩ C(R+) ∩ Liploc(R+), andalso in such a way that f satisfies (f1), (f4) and either (f2) or (f3) in its entiredomain R+; and finally so that Φ is locally bounded near v = 0. Here we assumethat c0 := F (v0) − vm

0 /m > 0. Clearly f verifies (62) with k1 = 0. As before,F (v) = c0 + vm/m for all v ≥ v0, and so

Φ(v) = c1 + c2vm, c1 := Npc0 > 0, c2 := p−N + Np/m < 0,

by construction and the fact that m < 0 and 1 maxd, v0 sufficiently large such that Φ(v) ≥ 0 for all v ≥ µ.

Let λ ∈ (0, 1). For all v ≥ µ/λ and all λ1, λ2 ∈ [λ, 1]

χ1(v) ∼ c1 ·Q(Cλλ

(1−m)/p2 · v(p−m)/p

)→∞ as v →∞,

since c1 > 0 by construction, namely (Φ1) is valid. While as v →∞

χ(v) ∼ c1 ·Q(cλλ

(m−1)/p2 · v(p+m)/p

)→

c1Q

(cλ/λ

(p+1)/p2

), if m = −p,

0, if m < −p,

that is (Φ) does not hold.

Letf(v) = e−v, v ≥ v0,

with v0 = 0 in case (f3). Otherwise we take v0 > 0 and define f in [0, v0] so thatf ∈ L1[0, v0]∩C(R+)∩Liploc(R+), F (v0) > 0, and also in such a way that f satisfies(f1), (f2) and (f4) in its entire domain R+, and Φ is locally bounded near v = 0.With the usual notation F (v) = c0− e−v for all v ≥ v0, with c0 = F (v0)+ e−v0 andeither F (v0) = 0 in case (f3) or F (v0) > 0 by construction in case (f2). Hence inboth cases c0 > 0. For all v ≥ v0

Φ(v) = c1 − [Np + (N − p)v]e−v, c1 = Npc0 > 0.


Therefore there is µ > maxd, v0 so large that Φ(v) ≥ 0 for all v ≥ µ. Fixλ ∈ (0, 1). Moreover for all λ2 ∈ [λ, 1] we have Cλv1/p′e(λ2v)/p → ∞ as v → ∞, sothat also for all λ1 ∈ [λ, 1]

χ1(v) = c1 − [Np + (N − p)λ1v]e−λ1v ·Q(Cλv1/p′eλ2v/p

)→∞

as v →∞, that is (Φ1) is valid. While cλ[vp+1e−λ2v]1/p → 0 as v →∞ and so alsoas v →∞

χ(v) = c1 − [Np + (N − p)λ1v]e−λ1v ·Q(cλ[vp+1e−λ2v]1/p

)→ 0,

since Q(0) = 0, that is (Φ) is not valid.

Finally we present two examples to which all Theorems 1–4 can be applied, sinceboth growth conditions (Φ) and (Φ1) hold.

Let −p < m < p∗Nf(v) = vm−1, v ≥ v0 > 0, (67)

and let f be defined in [0, v0] so that f ∈ L1[0, v0]∩C(R+)∩Liploc(R+), and also insuch a way that f satisfies (f1), (f4) and either (f2) or (f3) in its entire domain R+,Φ is locally bounded near v = 0, and with the further property that F (v0) > vm

0 /mif m 6= 0. Now set

c0(m) = F (v0)−

vm0 /m, if − p < m < p∗N , m 6= 0,

log v0, if m = 0,

consequently for v ≥ v0

F (v) = c0(m) +

vm/m, if − p < m < p∗N , m 6= 0,

log v, if m = 0,

and so, putting c2 := p−N + Np/m, we have for all v ≥ v0

Φ(v; m) = Npc0(m) +

c2v

m, if − p < m < p∗N , m 6= 0,

p−N + Np log v, if m = 0.

Therefore there is µ > maxd, v0 so large that Φ(v; m) ≥ 0 for all v ≥ µ. Indeedc2 > 0 when 0 < m < p∗N ; while Npc0(m) > 0 for −p < m < 0; and of courseΦ(v; 0) →∞ as v →∞ when m = 0.

Let λ ∈ (0, 1). For all λ1, λ2 ∈ [λ, 1] and v ≥ µ/λ

χ(v) = Φ(λ1v; m) ·Q(cλλ

(m−1)/p2 v(p+m)/p

)→∞ as v →∞.

Similarly for −p < m ≤ p

χ1(v) = Φ(λ1v; m) ·Q(Cλλ

(1−m)/p2 v(p−m)/p

)→∞ as v →∞.

While if p < m < p∗N there is ε > 0 sufficiently small that

m < p∗N − ε(m− p)N − p

. (68)

Now, for all λ1, λ2 ∈ [λ, 1] we have Cλλ(1−m)/p2 v(p−m)/p → 0 as v →∞, and so by

(65)1 for all v ≥ v1 ≥ µ/λ, with v1 large enough,

χ1(v) ≥ C1,ελm1 c2

[Cλλ

(1−m)/p2

]N+ε

· vm+(p−m)(N+ε)/p →∞


as v → ∞ since Npc0(m) > 0 by construction, c2 > 0 by the fact that m > p > 1and

m +(p−m)(N + ε)

p> 0

by (68). In conclusion also (Φ1) is valid.

Assume that (67) holds with m > −p and let f be defined in [0, v0] so thatf ∈ L1[0, v0] ∩ C(R+) ∩ Liploc(R+), and also in such a way that f satisfies (f1),(f4) and either (f2) or (f3) in its entire domain R+, that Φ is locally bounded nearv = 0, and with the further property that F (v0) > vm

0 /m if m 6= 0. Then repeatingthe same argument above, (Φ) and (Φ1) hold if and only if

m < p∗N .

Indeed the sufficient part is proved above. For the necessary part if m ≥ 0 then thepositivity of Φ forces that m < p∗N , while if m < 0 obviously there is nothing toprove. In this case all Theorems 1–4 can be applied.

Hence (Φ) and (Φ1) hold for (67) with m > −p if and only if m < p∗N , that isthey are subcritical growth conditions for f at ∞ in the sense of Sobolev embeddingwith weights.

Let 0 ≤ m < p∗N ,

f(v) = vm−1 log v, v ≥ v0 > 0, (69)

and let f be defined in [0, v0] so that f ∈ L1[0, v0] ∩ C(R+) ∩ Liploc(R+), and alsoin such a way that f satisfies (f1), (f4) and either (f2) or (f3) in its entire domainR+, and Φ is locally bounded near v = 0. As before, put

c0(m) = F (v0) +

vm0 [1−m log v0]/m2, if 0 < m < p∗N ,

−log2 v0/2, if m = 0,

and so for all v ≥ v0

F (v) = c0(m) +

vm[m log v − 1]/m2, if 0 < m < p∗N ,

log2 v/2, if m = 0,

Φ(v; m) = Npc0(m)+

vm[m(N − p)(p∗N −m) log v −Np]/m2, if 0 < m < p∗N ,

log v[Np log v − 2(N − p)]/2, if m = 0.

Hence there is µ > maxd, v0 so large that Φ(v) ≥ 0 for all v ≥ µ, since m < p∗N .Let λ ∈ (0, 1). For all λ1, λ2 ∈ [λ, 1] and v ≥ µ/λ we have

χ(v) = Φ(λ1v; m) ·Q(cλ[λm−1

2 vm+p log(λ2v)]1/p)→∞ as v →∞.

Namely (Φ) holds.Now to prove the validity of (Φ1), we distinguish two cases again for all λ1,

λ2 ∈ [λ, 1] and v ≥ µ/λ. If m ∈ [0, p), then

χ1(v) = Φ(λ1v; m)Q

(Cλ

[λ1−m

2 vp−m · 1log(λ2v)

]1/p)→∞ as v →∞.

While in the remaining case m ∈ [p, p∗N ), we can argue as for the example (67),since

Cλ

[λ1−m

2 vp−m

/log(λ2v)

]1/p

→ 0


as v →∞. Therefore, taking ε > 0 so small that

m +(p−m)(N + ε)

p> 0,

by (65)1 for v sufficiently large we get

χ1(v) ≥ C1,εΦ(λ1v;m)[Cλλ(1−m)/p2 ]N+ε

[vp−m

log(λ2v)

](N+ε)/p

∼ C1,ε[Cλλ(1−m)/p2 ]N+ε(p−N)(1− p∗N/m)λm

1 vm+(p−m)(N+ε)/p

· log(λ1v)

[log(λ2v)](N+ε)/p,

and (Φ1) follows at once letting v →∞.

6. Preliminary Lemmas for the existence of radial ground states. Through-out the section we assume that the non–linearity f in (25) verifies assumptions (f1),(f2) and (f4). Let

I+ := α ≥ β : `α > 0. (70)Of course I+ and I− are disjoint, where I− is given in (32). We shall prove belowsome properties useful for the proof of the main existence Theorems 7 and 10.

Lemma 6. β belongs to I+.

Proof. Let v be a solution of

[q(t)|v′(t)|p−1]′ = q(t)f(v), v(0) = β, v′(0) = 0,

defined in Iβ = (0, tβ), given in (24). From (26), we get

E(0) =ρp(0)

p′+ F (v(0)) = F (β) = 0, (71)

thanks to (f2). Moreover, E is strictly decreasing in Iβ by (27), hence E(t) < 0 inIβ by (71). Now, fix t0 ∈ Iβ , then

F (v(t)) ≤ E(t) < E(t0) < 0 in (t0, tβ).

Hence, by letting t → tβ , we get F (`β) ≤ E(t0) < 0, where `β is defined in (28). Inturn `β > 0 by (f2), since F (0) = 0 by (f1). Thus β ∈ I+.

Lemma 7. I+ is open in [β,∞).

Proof. Fix α∗ ∈ I+. Let v∗ be the solution of (25), with α replaced by α∗, definedin its maximal interval I∗ = (0, t∗), in the sense of (24). Clearly

`∗ ∈ (0, β) and v′∗(t) → 0 as t → t−∗ (72)

by Lemma 3 (i), (ii) and the fact that α∗ ∈ I+. Thus

limt→t−∗

E∗(t) = F (`∗) < 0,

by (26) and (f2), since `∗ ∈ (0, β).First fix t0 ∈ I∗ such that E∗(t0) < 0. If α > 0 is chosen sufficiently close to

α∗ and v is the corresponding solution of (25), then by Lemma 2 we have that(0, t0] ⊂ I, where I is the maximal interval of continuation of v. FurthermoreE(t0) ≤ E∗(t0)/2 < 0. As in the proof of Lemma 6, this implies that α is in I+.

Lemma 8. I− is open.


Proof. Fix α ∈ I− and let (αk)k∈N be any sequence of positive numbers convergentto α. Let v be the solution of (25), with v(0) = α, in its maximal domain ofcontinuation Iα = (0, tα); and similarly denote by vk the solution of (25), withvk(0) = αk, in the maximal domain Iαk

= (0, tk), in the sense of (24). Denoteby E and Ek the energy functions along the solutions v and vk, respectively. Putc := E(tα)/2. Clearly c > 0 by (26), since v′(tα) < 0 and `α = 0 by the factthat α ∈ I−. By Lemma 2 of course ‖Ek − E‖∞ → 0 as k → ∞, so that there iss0 ∈ (tα/2, tα) such that 2c < E(s0) < 3c by (27), and for all k ∈ N sufficientlylarge

tk > s0, c ≤ Ek(s0) ≤ 4c, vk(s0) ≤ 2v(s0) ≤ β. (73)

By (27), integrating on [s0, tk), we have by (q2)

|Ek(tk)− Ek(s0)| ≤ q′(s0)q(s0)

· sup[s0,tk)

ρp−1k (s) ·

∫ tk

s0

ρk(s)ds,

where ρk := |v′k|. Now∫ tk

s0

ρk(t)dt =∫ tk

s0

−v′k(t)dt =∫ vk(s0)

vk(tk)

dv ≤ vk(s0),

since v′k < 0 in Ik by (24). Hence for all k sufficiently large

|Ek(tk)− Ek(s0)| ≤ q′(s0)q(s0)

vk(s0) · sup[s0,tk)

ρp−1k (t) ≤ M0 sup

[s0,tk)

ρp−1k (t), (74)

where M0 := 2q′(s0)v(s0)/q(s0). By (27) and (73)2 for all t ∈ [s0, tk)

ρpk(t) = p′[Ek(t)− F (vk(t))] ≤ p′[Ek(s0)− F (vk(t))] ≤ p′[4c + F ],

where F := maxv∈[0,β]

|F (v)|. In turn

sup[s0,tk)

ρp−1k (t) ≤ [p′(4c + F )]1/p′ := c,

and by (74) we obtain|Ek(tk)− Ek(s0)| ≤ cM0. (75)

Clearly (75) remains valid when s0 is replaced by any t in (s0, tα) ⊂ (tα/2, tα).Since v(t) → 0 as t → tα− , being α ∈ I−, then Ek(s0) ≥ c > 0 by (73)2. Moreover,since F (vk(tk)) ≤ 0, being vk(tk)) ≤ β, then ρp

k(tk) > 0 by (26), that is

v′k(tk) < 0, tk < ∞, vk(tk) = 0,

by Lemma 3 (iii) and (24). Hence αk ∈ I− for all k sufficiently large, namely I− isopen.

7. Existence results. In this section we shall establish existence of radial solutionsof (1), assuming in the sequel that 1 0.

Theorem 5. Assume (f1), (f3)–(f5) and (Φ), with γ = ∞. Then (1) admits asemi–classical non–singular radial crossing solution uα in the ball Bα, with

uα(0) = α > 0, Duα(0) = 0, rα < ∞,

uα(x) = 0 and Duα(x) · x < 0 for x ∈ ∂Bα.(76)


Proof. By virtue of Theorem 1 there is α ∈ I− and a corresponding solution v of (7)in the interval Iα = (0, tα), maximal in the sense of (24). Hence uα(x) = vα(t(|x|))in Bα, rα = r(tα) by (12), is the required solution.

Remarks. Theorem 5 can be applied to all equations (4), including the classicalMatukuma equation, when f verifies (f5) and γ = ∞.

In the recent paper [9] some existence, non–existence and uniqueness results forradial ground states of some special cases of (2) are given when f > 0 everywherein R+ but singular at u = 0. Moreover in [9] it is required that g ≡ 1, and h iscontinuous also at r = 0 and verifies an integral condition. In their prototype

f(u) = u−m + u℘, (77)

the main existence Theorem 1.1 of [9] can be applied provided that 1 0 if 1 < p ≤ 2, or when(p − 2)n + `(p − 1) + p < 0 if p ≥ 2, that is in both cases when ` < 0, so that hmust be singular at r = 0. Hence Theorem 1.1 of [9] cannot be applied in this case,since in [9] the weight h is required to be continuous also at r = 0. In the famousMatukuma case, namely when g ≡ 1 and h(r) = (1 + r2)−1, n = 3, p = 2, themain condition (1.4) of [9] again fails to hold, so that Theorem 1.1 of [9] cannot beapplied.

For (77) conditions (f1), (f3)–(f5) and both (Φ) and (Φ1) hold provided that

1 < p < N, 1− p < m < 1, −m < ℘ < p∗N − 1.

Clearly Theorem 5 can be applied in both examples discussed above, and actuallyalso in the generalized Matukuma equations (see (1.4) of [17]), namely when g ≡ 1and h(r) = rσ−p′/(1 + rp′)σ/p′ , σ > 0, and

n ≥ 2, p ≥ 2, with N = pn + σ − p′

p + σ − p′> 1.

Hence in the classical subcase p = 2 and n = 3 it results N = 2(1 + σ)/σ > 2. Inany case here 1 < p < N if and only if 1 < p < n.

We now present another result when γ is possibly finite.

Theorem 6. Assume that (f1), (f3), (f4), (q4) and (Φ) hold. Then (1) admitsa semi–classical non–singular radial crossing solution uα in the ball Bα, satisfying(76).

Proof. Here by virtue of Theorem 2 there is α ∈ I−, so that 0 < α < γ, and acorresponding solution vα of (25) in the interval Iα = (0, tα), maximal in the senseof (24). Hence uα(x) = vα(t(|x|)) in Bα, rα = r(tα) by (12), is the required solutionalso in this case.

Remarks. When f is of the required type, Theorem 6 can be applied to all equa-tions (4), with exponents verifying (5) and either (57) or (58), but not, for instance,to the Matukuma equation.

Some existence and non–existence results for radial ground states of special casesof (2) are given in [5] in the case in which f is non–negative for u > 0 small. Theyprove existence of non–trivial positive radial solutions of (1) in the interesting casein which the continuous non–linearity f may depend on r, but is continuous also atu = 0. Furthermore, they consider (2) when g(r) = rk, k +n−p > 0, h(r) ∼ r`h(r)


as r →∞ with `− k + 1 > 0 and f(r, u) ≥ h(r)um for u > 0 sufficiently small andr sufficiently large, with m > p− 1, and h verifying some integral conditions.

As a corollary of Theorem 6 we obtain existence of crossing solutions for thep–Hessian operator Hp−1, p ∈ Z, according to the notation in [23] and [5], see also[17]. For instance, in the first prototype studied in Theorem 5.2 of [5], that is

div(r2−p|Du|p−2Du) +|u|m−1u

1 + rσ= 0, r > 0, p > 1,

u(0) = α > 0, u′(0) = 0,

(78)

existence of crossing solutions is proved by Theorem 6, provided that

0 ≤ σ < 2(p− 1) < n, −1 < m <(p− 1)(n + 2)n− 2(p− 1)

. (79)

Moreover, when (79) holds with σ = 0, p ≥ 2 and m > 1, Theorem 5.2–(M1) of[5] can also be applied so that problem (78) has no positive solutions. ThereforeTheorem 6 adds more information in the mentioned non–existence result of [5]. Forthe same reason, also in the case σ > 0, p ≥ 2 and m > 1 Theorem 6 adds furtherinformation with respect to Theorem 5.2–(M4) of [5] since

(p− 1)(n + 2)n− 2(p− 1)

>(n− σ)p− [n− 2(p− 1)]

n− 2(p− 1)and p ≤ 2(p− 1).

In the other special case treated in [5], namely Batt–Faltenbacher–Horst case, thatis when (4) reduces to

div(r2−p|Du|p−2Du) +rσ−p

(1 + rp)σ/p|u|m−1u = 0, r > 0, p > 1,

u(0) = α > 0, u′(0) = 0,

(80)

Theorem 6 shows that problem (80) admits crossing solutions when

p > 2, σ > 0, n > 2(p− 1), −1 < m <(n + σ − p)p− n + 2(p− 1)

n− 2(p− 1),

while in the somewhat complementary case

2 1, n > 2(p− 1), m ≥ (n + σ − p)p− n + 2(p− 1)n− 2(p− 1)

,

Theorem 5.3 of [5] guarantees that (80) has a ground state solution, so that nowthe picture is more detailed.

We now turn to the more delicate case in which the non–linearities treated areof the type (f2).

Theorem 7. Assume that (f1), (f2), (f4), (q4) and (Φ) hold.If f is continuous also at u = 0 and f(0) = 0, then (1) admits a semi–classical

non–singular radial ground state uα, with β < uα(0) = α < γ, which is compactlysupported in Rn if ∫

0+

du

|F (u)|1/p< ∞ (81)

otherwise positive if

|F (u)| ≤ G(u), u ∈ [0, ε), ε > 0,

∫

0+

du

|G(u)|1/p= ∞, (82)


where G : [0, ε) → R, with G(0) = 0, is non–decreasing. When (81) holds, then thecompactly supported solution uα is also a semi–classical non–singular radial solutionof the free boundary problem (3) for some R > 0.

When lim supu→0+ f(u) < 0, then problem (1) admits no semi–classical non–singular radial ground states, while (3) has a semi–classical non–singular radialsolution for some R > 0.

When f is singular at u = 0 and lim supu→0+ f(u) ≥ 0, then either (1) has apositive semi–classical non–singular radial ground state or (3) has a radial solutionfor some R > 0. If (82) is satisfied, then the first case occurs; if (81) is satisfied,the second case occurs.

Moreover, if for some ν < 1

g(r)h(r)

∼ c rν as r → 0+, c > 0, (83)

then the solution u is regular, that is Du is Holder continuous at x = 0, withDu(0) = 0; while if ν ∈ [1, p), then u is Holder continuous at x = 0. In both cases

u ∈ W 1,ploc (Rn),

when also 1 < p ≤ n.

Proof. As already noted, I+ and I− are disjoint. By Theorem 2 and Lemmas 6–8,the sets I+ and I− are open and not empty. Hence there is α /∈ I+ ∪ I−, whosecorresponding solution vα of (25) is positive in the maximal interval Iα, given by(24). Since α /∈ I+, then `α = 0 by (70). Since α /∈ I−, then either tα = ∞ ortα < ∞. In both cases v′α(tα) = 0 by Lemma 3 (iii) and the fact that α /∈ I−.In the first case vα is a positive semi–classical radial ground state of (25) and souα(x) = vα(t(|x|)) is a semi–classical radial ground state of (1). In the second caseuα(x) is a solution of (3) with R = Rα = r(tα). In particular in this latter casewhen f is continuous also at u = 0, with f(0) = 0, the solution, when it is extendedto all x, with |x| > Rα, by the value 0, becomes a compactly supported radialground state of (1).

In conclusion we have shown that, if f is continuous also at u = 0, with f(0) = 0,problem (1) admits a semi–classical non–singular radial ground state uα, with β <uα(0) = α < γ, which is compactly supported in Rn if (81) holds or everywherepositive in Rn if (82) is valid by virtue of Theorem 5.7 of [17].

Now, when lim supu→0+ f(u) < 0, if there would exist a semi–classical non–singular radial ground state u of (1) or equivalently a semi–classical radial groundstate v of (25), then by Lemma 3

|v′(t)|p−1 → 0 as t →∞, (84)

so that by (31) and (q2)lim sup

t→∞

[|v′(t)|p−1]′

< 0.

This is clearly impossible, since by the mean value theorem and by (84) there wouldexist a sequence (tk)k tending to ∞, along which [|v′(tk)|p−1]′ → 0 as k →∞. Thiscontradiction proves the claim since the only possibility is that Rα < ∞. Hencethere exists a semi–classical non–singular solution of (3).

Combining now the above conclusions with Theorem 5.7 of [17] we immediatelyget the assertion in the case in which lim supu→0+ f(u) ≥ 0 and f is singular atu = 0.


Finally, the regularity result valid under (83) is a direct application of Corol-lary 3.3 of [17].

For a more general discussion of the validity of the strong maximum and compactsupport principles for solutions, radial or not, of quasilinear elliptic inequalities, aswell as on applications of these principles to variational problems on manifolds andto existence of radial dead cores, we refer to [20].

As shown in Section 4, the above results continue to hold when assumption (Φ)is replaced by (Φ1). More precisely

Theorem 8. Assume (f1), (f3)–(f5) and (Φ1). Then (1) admits a semi–classicalradial non–singular crossing solution uα in the ball Bα, satisfying (76).

Proof. To do this we can repeat the proof of Theorem 5 word for word with thesingle exception that Theorem 1 is replaced by Theorem 3.

Theorem 9. Assume that (f1), (f3), (f4), (q4) and (Φ1) hold. Then (1) admitsa semi–classical non–singular radial crossing solution uα in the ball Bα, satisfying(76).

Proof. This result can be shown following the proof of Theorem 6 word for wordwith the exception that Theorem 2 is replaced by Theorem 4.

Theorem 10. Assume the validity of (f1), (f2), (f4), (q4) and (Φ1). Then theconclusions of Theorem 7 continue to hold.

Proof. To do this we can repeat the proof of Theorem 7 word for word with theexception that Theorem 2 is replaced by Theorems 3 and 4.

8. Non–existence results. In this section a non–existence result for positive ra-dial semi–classical non–singular ground states of (1) is established via essentiallythe technique of Theorem 3.2 of [13]. We introduce(f6) there exists β′ > 0 such that f(u) ≤ 0 for 0 < u < β′.Of course when both (f2) and (f6) hold, then β′ < β.

Theorem 11. Assume (A1)–(A4), (f1), (f2) and (f6), and that 1 0 is sufficiently small.Finally, if

(A6)[1p· a′

a+

1p′· b′

b

] ∫ r

0b(s)ds

b(r)≥ N − 1

Nin R+,

then problem (1) does not admit any positive radial semi-classical non-singularground state.

Proof. Assume by contradiction that there exists a semi–classical non–singular po-sitive radial ground state u of (1), and denote by v(t) = u(r(t)) the correspondingclassical ground state of (7). By (29) and (7) we get that

[Q(t)E(t) + κq(t)|v′(t)|p−2v′(t)v(t)]′ = qE − Qq′

q|v′|p + κq|v′|p − κqvf(v).


Using (26), by integration of the above equality we obtain

Q(t)E(t) + κq(t)|v′(t)|p−2v′(t)v(t)

=∫ t

0

q

F (v)− κvf(v) +

(κ− Qq′

q2+

1p′

)|v′|p

ds,

(87)

for all t > 0 and κ ∈ R. Now take κ = 1/(℘ + 1), where ℘ is given in (85). Then by(85) and the fact that (A6) is equivalent to (q5), we obtain in R+

κ− Qq′

q2+

1p′≤ 1

℘ + 1− 1

p∗N≤ 0.

Hence by (86) the right side of (87) is strictly negative. In the left hand side of (87)we have that

limt→∞

q(t)|v′(t)|p−2v′(t)v(t) = 0

by Proposition 6.1 of [17], since v is a ground state of (7). On the other hand for tsufficiently large

[Q(t)E(t)]′ ≤ q(t)|v′(t)|p[

1p′− Qq′

q2

]≤ −q(t)

p∗N|v′(t)|p < 0 (88)

by (26), (27), the fact that F (v) < 0 if v is small by (f2) and (q5). Furthermoreby Proposition 6.1 of [17] we have that lim inft→∞Q(t)E(t) = 0, hence (88) impliesthat

limt→∞

Q(t)E(t) = 0.

Thus the left hand side of (87) tends to 0 as t →∞ and so∫ ∞

0

q(s)[F (v(s))− κv(s)f(v(s))]ds = −∫ ∞

0

q(s)(

κ− Qq′

q2+

1p′

)|v′(s)|pds.

Now, since κ = 1/(℘ + 1) as above, we get the required contradiction since the leftside is strictly negative by (86) while the right hand side is non–negative.

As a consequence of the main Theorems 10 and 11 we prove the corollary of theIntroduction.

Proof of Corollary 1. Clearly f in (8) satisfies (f1), (f2), (f4) and (f6), with

β =(

℘ + 1m + 1

)1/(℘−m)

, β′ = 1 < β, γ = ∞,

since −1 < m < ℘. Moreover when 1 < p < N the principal conditions (Φ) and(Φ1) are valid if ℘ < p∗N − 1.

The existence part of the corollary follows from either Theorem 7 or Theorem 9,while the non–existence result from Theorem 11. When Iα = R+, then (31) admitsa non–negative ground state only when lim supu→0+ f(u) ≥ 0, that is in case (8)when limu→0+ f(u) = 0, namely when m > 0. Furthermore by Corollary 5.8 of [17]a ground state of (7), with f given by (8), is positive in the entire Rn if and onlyif m ≥ p− 1, and so the claim is proved.

Finally, the regularity property under (83) follows exactly as in the proof ofTheorem 7. ¤


Remarks. 1. As noted by Ni and Serrin in the Remark at the end of Theorem 3.3of [13], if f(u) = u℘ϕ(u), ϕ ∈ C1(R+), ϕ(u) < 0 for u small, then

(℘ + 1)F (u)− uf(u) = −∫ u

0

t℘+1ϕ′(t)dt. (89)

Consequently, condition (86) holds if ϕ′(u) ≥ 0 in R+, and ϕ′(u) > 0 for u > 0sufficiently small.

If, for example, we consider f(u) = −um + u℘, −1 < m < ℘, we see thatϕ(u) = 1− um−℘ and

(℘ + 1)F (u)− uf(u) =m− ℘

m + 1um+1.

Thus (86) holds since m < ℘.2. As used in the proof of Theorem 11, assumption (A6), in terms of the radial

weights a and b of the original radial equation (2), is equivalent to condition (q5) interms of the new weighted equation (7), where q is given by (13), since Q(t(r)) =∫ r

0b(s)ds as noted in the Remark before Theorem 3.In particular when g ≡ 1 assumption (A6) becomes

(A6)′[n− 1

r+

1p′· h′

h

] ∫ r

0sn−1h(s)ds

rn−1h(r)≥ N − 1

N.

Finally, we give some examples of functions g and h, for which (A1)–(A6) or (q1)–(q5) hold.

3. Of course in the case

g(r) ≡ 1, h(r) = r`,

then as noted in Section 2 the main structure assumptions (A1)–(A4) hold if (19)is satisfied, with N > 1 given in (20). In this case q(t) = tN−1 and so (q4) and (q5)trivially hold. In conclusion all (A1)–(A6) are valid.

Another interesting example is given by

g(r) ≡ 1, h(r) = log(1 + r),

where (A1)–(A5) hold, with N = p(n + 1)/(p + 1) > 1. Furthermore, when p = 2and n = 3, it is not hard to see that also (A6) is valid.

Moreover another example is given by

g(r) = h(r) = r1−n(e√

r − 1), q(t) = e√

t − 1, r, t ∈ R+0 .

Indeed (q1), (q2) and (q4) trivially hold, q(0) = 0, (q3) is satisfied with N = 3/2.Finally (q5) is verified when n = 3 and p = 2, since Qq′/q2 is an increasing functionsuch that limt→0+ Qq′/q2 = 1/3(= (N − 1)/N) and limt→∞Qq′/q2 = 1.

4. Actually Theorem 11 is the special case c = 1 − 1/N of the following moregeneral non–existence result:

Assume (A1)–(A4), (f1), (f2) and (f6), and that 1 0 is sufficiently small, then problem (1) does notadmit any positive radial semi–classical non–singular ground state.

The proof is exactly the same as that of Theorem 11, where now c replaces theprevious main number 1− 1/N . Note that here ℘ + 1 = p′/(p′c− 1) ≥ p∗N , and c isany positive number, such that

1N≤ 1− c <

1p.

5. The function

f(u) = −c1um + c2u

℘, −1 < m < ℘, (90)

where c1, c2, are positive constants, can be transformed by the change of variableu = ηv, η = (c1/c2)1/(℘−m) to the form

f(v) = c(−vm + v℘), c = c1ηm = c2η

℘ > 0.

Hence (f1), (f2), (f4) and (f6) are also satisfied by (90), since −1 < m < ℘, withγ = ∞.

The study of uniqueness of radial ground states of (1) or non singular solutionssemi–classical of (3) is very delicate. The first results are contained in [17], wherethe non–linearity f is assumed to have a sublinear growth at infinity, see also relatedresults, even if more specific, given in [9]. In particular in [17] it was proved forthe special non–linearity (89) that the corresponding equation (1) admits at mostone semi–classical radial ground state u, with 0 < u(0) < γ = ∞, when (6) holds,namely when

p ≥ 2, −1 < m < ℘ ≤ p− 1, m ≤ 1 +p− 3p− 1

℘.

Clearly (6) allows values m > 0 and ℘ < 0, though not both at the same time. Inthis case, when also condition (q4) holds, both Theorem 8.4 of [17] and Corollary 1can be applied to (1), that is (1) admits one and only one semi–classical radialground state u, with u(0) > 0, when f satisfies (6).

Acknowledgements. The authors were supported by MIUR project “Metodi Va-riazionali ed Equazioni Differenziali non Lineari”.

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Received April 2005; revised November 2005.E-mail address: [email protected]

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