COMMUNICATIONS ON doi:10.3934/cpaa.2013.12.1243PURE AND APPLIED ANALYSISVolume 12, Number 3, May 2013 pp. 1243–1257

INFINITE MULTIPLICITY FOR AN INHOMOGENEOUS

SUPERCRITICAL PROBLEM IN ENTIRE SPACE

Liping Wang

Department of mathematics, East China Normal University,500 Dong Chuan Road, Shanghai, China, 200241

Juncheng Wei

Department of mathematics, The Chinese University of Hong Kong,Shatin, New Territories, Hong Kong

(Communicated by Hongjie Dong)

Abstract. Let K(x) be a positive function in RN , N ≥ 3 and satisfy

lim|x|→∞

K(x) = K∞ where K∞ is a positive constant. When p > N+1N−3

, N ≥ 4,

we prove the existence of infinitely many positive solutions to the followingsupercritical problem:

∆u(x) +K(x)up = 0, u > 0 in RN , lim|x|→∞

u(x) = 0.

If in addition we have, for instance, lim|x|→∞

|x|µ(K(x)−K∞) = C0 6= 0,

0 < µ ≤ N − 2p+2p−1

, then this result still holds provided that p > N+2N−2

.

1. Introduction and Statement of the results. The purpose of this paper isto establish the existence of infinitely many positive solutions to the followinginhomogeneous equation∆u+K(x)up = 0,

u > 0 in RN , lim|x|→∞

u(x) = 0. (1)

where N ≥ 3, p > N+2N−2 and 0 < a ≤ K(x) ≤ b < +∞.

Semilinear elliptic equations like above seem to arise naturally in many appliedareas. We refer the interested readers to [2], [3] and [7] for a brief history andbackground of (1).

In [6], Ding and Ni showed that for p ≥ N+2N−2 , x · ∇K(x) ≤ 0 and K(x) is

symmetric in xj , j = 1, ..., N , then equation (1) admits infinitely many solutions.Using sub-super solution method, Gui [7]-[8] showed that there exists an exponentpc (defined at (6) below) such that for p ≥ pc, N ≥ 11, equation (1) has infinitelymany (well-separated) solutions in the case when K is radially symmetric. Recentextensions can be found in Bae and Ni [3], and Bae [1]. However, in [1], [3], [7]and [8], it is always assumed that p ≥ pc and N ≥ 11. The case of N ≤ 10 and

2000 Mathematics Subject Classification. Primary 35B35, 92C15; Secondary 35B40, 92D25.Key words and phrases. Supercritical, infinite multiplicity, entire space.

1243

1244 LIPING WANG AND JUNCHENG WEI

N+2N−2 < p < pc has been left open. Note that in this case, the method of sub-

super solution breaks down. Other related results can be found in Wang-Wei [11],Yanagida-Yotsutani [12].

In this paper, under reasonable conditions onK, we establish that when p > N+2N−2 ,

equation (1) has a continuum of solutions. Our basic assumption is the following

(H) K(x) is smooth, lim|x|→∞

K(x) = K∞ > 0.

Our main result is the following:

Theorem 1.1. Assume that K(x) satisfies (H) and p > N+1N−3 , N ≥ 4. Then problem

(1) has a continuum of solutions uλ(x) (parameterized by λ ≤ λ0) such that

limλ→0

uλ(x) = 0

uniformly in RN .The same result holds when N+2

N−2 < p ≤ N+1N−3 provided that K is symmetric with

respect to N coordinate axis, namely

K(x1, . . . , xi, . . . , xN ) = K(x1, . . . ,−xi, . . . , xN ), for all i = 1, . . . , N.

The basic obstruction to extend the result to the whole supercritical range isthat the linearized operator around canonical approximation will no longer be ontoif N+2N−2 < p ≤ N+1

N−3 . This problem can be overcome through a further condition on

K(x). We have the validity of the following result.

Theorem 1.2. Assume that K(x) satisfies (H) and N+2N−2 < p ≤ N+1

N−3 if N ≥ 4,

p > N+2N−2 if N = 3. Then the result of Theorem 1.1 also holds true if

(a) there exists µ > N , such that∫RN

(K(x)−K∞

)6= 0, |K(x)−K∞| ≤ C|x|−µ, |x| ≥ 1;

or(b) there exist a bounded function f : SN−1 → R and N − 2p+2

p−1 < µ ≤ N such

that

lim|x|→∞

(|x|µ(K(x)−K∞)− f(

x

|x|))

= 0,

where f(x) satisfies∫RN f( x

|x| )|x|−µ|x+ω|−

2(p+1)p−1 6= 0 for any ω ∈ ∂B1(0) if µ < N

and∫SN−1 f 6= 0 if µ = N .

Instead of using sub-super solution method (which limits the applicability on theexponent p), we use asymptotic analysis and Liapunov-Schmidt reduction methodto prove Theorem 1.1 and Theorem 1.2. This is based on the construction of asufficiently good approximation and asymptotic analysis. It is well known that theproblem

∆W +W p = 0 in RN (2)

possesses a positive radially symmetric solution W (|x|) whenever p > N+2N−2 . We fix

in what follows the solution W of (2) such that

W (0) = 1. (3)

Then all radial solutions to this problem can be expressed as

Wλ(x) = λ2p−1W (λx). (4)

INFINITE MULTIPLICITY 1245

At main order one has

W (r) ∼ Cp,Nr−2p−1 as r →∞, (5)

which implies that this behavior is actually common to all solutions Wλ(x). In [10]and [9], it is shown that if p = pc,

W (r) =β

1p−1

r2p−1

+a1 log r

rµ0+ o(

log r

rµ0), r →∞,

where β = 2p−1 (N − 2− 2

p−1 ), a1 < 0, µ0 >2p−1 and if p > pc

W (r) =β

1p−1

r2p−1

+a1rµ0

+ o(1

rµ0), r →∞,

where

pc =

{(N−2)2−4N+8

√N−1

(N−2)(N−10) , N > 10,

∞, N ≤ 10.(6)

The idea is to consider Wλ(x) as an approximation for a solution of (1), providedthat λ > 0 is chosen small enough. To this end, we need to study the solvability ofthe operator ∆+pW p−1 in suitable weighted Sobolev space. Recently, this issue hasbeen studied in Davila-del Pino-Musso [4] and Davila-del Pino-Musso-Wei [5]. Inparticular, our method here is closely related to [5] where standing wave solutionsare constructed for nonlinear Schrodinger equations

∆u− V (x)u+ up = 0, u > 0 in RN , lim|x|→+∞

u(x) = 0 (7)

with

V (x) = o(1

|x|2), V (x) ≥ 0. (8)

Throughout the paper, the symbol C denotes always a positive constant inde-pendent of λ, which could be changed from one line to another. Denote A ∼ B ifand only if there exist two positive numbers a, b such that aA ≤ B ≤ bA.

2. The solvability of linearized operator ∆ + pW p−1. Our main concern inthis section is to study the existence of solution in certain weighted spaces for

∆φ+ pW p−1φ = h in RN , (9)

where W is the radial solution to (2), (3) and h is a known function having a specificdecay at infinity.

We work in weighted L∞ spaces adjusted to the nonlinear problem (1) and inparticular take into account the behavior of W at infinity. We are looking for asolution φ to (9) that is small compared to W at infinity, thus it is natural to

require that it has a decay of the form O(|x|−2p−1 ) as |x| → +∞. As a result

we shall assume that h behaves like this but with two powers subtracted, that is,

h = O(|x|−2p−1−2) at infinity. These remarks motivate the definitions

‖φ‖∗ = sup|x|≤1

|x|σ|φ(x)|+ sup|x|≥1

|x|2p−1 |φ(x)|, (10)

and

‖h‖∗∗ = sup|x|≤1

|x|2+σ|h(x)|+ sup|x|≥1

|x|2p−1+2|h(x)|, (11)

where σ > 0 will be fixed later as needed.

1246 LIPING WANG AND JUNCHENG WEI

The following lemmas and remark on the solvability are due to Davila-del Pino-Musso [4], Davila-del Pino-Musso-Wei [5]:

Lemma 2.1. Assume that p > N+1N−3 , N ≥ 4. For 0 < σ < N − 2 there exists a

constant C > 0 such that for any h with ‖h‖∗∗ < ∞, equation (9) has a solutionφ = T (h) such that T defines a linear map and

‖T (h)‖∗ ≤ C‖h‖∗∗.

An obstruction arises if N+2N−2 < p ≤ N+1

N−3 , which can be handled by consideringsuitable orthogonality conditions with respect to translations of W . Let us define

Zi = η∂W

∂xi(12)

and η ∈ C∞0 (RN ), 0 ≤ η ≤ 1,

η(x) = 1 for |x| ≤ R0, η(x) = 0 for |x| ≥ R0 + 1.

We work with R0 > 0 fixed large enough.Then we have

Lemma 2.2. Assume N ≥ 3, N+2N−2 < p < N+1

N−3 and let 0 < σ < N − 2. There is a

linear map (φ, c1, . . . , cN ) = T (h) defined whenever ‖h‖∗∗ <∞ such that

∆φ+ pW p−1φ = h+

N∑i=1

ciZi in RN (13)

and

‖φ‖∗ +

N∑i=1

|ci| ≤ C‖h‖∗∗.

Moreover, ci = 0 for all 1 ≤ i ≤ N if and only if h satisfies∫RN

h∂W

∂xi= 0 ∀ 1 ≤ i ≤ N. (14)

Remark 1. If p = N+1N−3 , the conclusion of Lemma 2.2 still holds if one redefines

the norms as

‖φ‖∗ = sup|x|≤1

|x|σ|φ(x)|+ sup|x|≥1

|x|2p−1+α|φ(x)|,

‖h‖∗∗ = sup|x|≤1

|x|σ+2|h(x)|+ sup|x|≥1

|x|2p−1+α+2|h(x)|,

where α > 0 is fixed small.

Although we have got the solvability of the corresponding linear operator, we doneed lots of careful analysis to get true solutions by fixed point theorem and degreetheory. Especially for the case N+2

N−2 < p ≤ N+1N−3 , more computations are necessary

to guarantee that (14) holds.

INFINITE MULTIPLICITY 1247

3. The proof of Theorem 1.1. Let p > N+1N−3 . We prove Theorem 1.1 in this

section. The main idea is to use Lemma 2.1 and a contraction mapping principle.

Proof. By a change of variables K1p−1∞ λ−

2p−1u(xλ ), equation(1) is equivalent to

∆u+ up + K(x

λ)up = 0 in RN , (15)

where

K(x

λ) =

K(xλ )

K∞− 1. (16)

Note that by our assumption on K, for any fixed x 6= 0, we have

K(x

λ) = o(1).

We look for a solution of (15) of the form u = W + φ, which yields the followingequation for φ

∆φ+ pW p−1φ = N(φ)− K(x

λ)(W + φ)p,

whereN(φ) = −(W + φ)p +W p + pW p−1φ. (17)

Using the operator T defined in Lemma 2.1 we are led to solving the fixed pointproblem

φ = T

(N(φ)− K(

x

λ)(W + φ)p

). (18)

We use a fixed-point argument: Consider the set

F = {φ : RN → R | ‖φ‖∗ ≤ ρ},

where ρ ∈ (0, 1) is to be chosen and the operator A(φ) = T(N(φ)−K(xλ )(W+φ)p

).

We now prove that A has a fixed point in F.For any φ ∈ F, by the arguments in [4]-[5], we know that for 0 < σ < 2

p−1 chosen

in (10), (11), it holds‖N(φ)‖∗∗ ≤ C(‖φ‖2∗ + ‖φ‖p∗). (19)

Next we estimate ‖K(xλ )(W + φ)p‖∗∗. Let R > 0. Observe that

sup|x|≤1

|x|2+σ|K(x

λ)(W + φ)p| ≤C sup

|x|≤1|x|2+σ|K(

x

λ)|(‖W‖p∞ + |φ|p)

≤C sup|x|≤λR

· · ·+ C supλR≤|x|≤1

· · ·

Butsup|x|≤λR

|x|2+σ|K(x

λ)|(‖W‖p∞ + |φ|p) ≤ C(λR)2+σ + C‖φ‖p∗, (20)

supλR≤|x|≤1

|x|2+σ|K(x

λ)|(‖W‖p∞ + |φ|p) ≤ Ca(R)(1 + ‖φ‖p∗) ≤ Ca(R), (21)

wherea(R) = sup

|x|≥R|K(x)|, then lim

R→∞a(R) = 0. (22)

On the other hand,

sup|x|≥1

|x|2+2p−1 |K(

x

λ)(W + φ)p| ≤Ca(

1

λ) sup|x|≥1

|x|2+2p−1 (|W |p + |φ|p)

≤Ca(1

λ)(1 + ‖φ‖p∗) ≤ Ca(

1

λ).

(23)

1248 LIPING WANG AND JUNCHENG WEI

Thus by (20), (21), (23), we get

‖K(x

λ)(W + φ)p‖∗∗ ≤ C

(a(R) + a(

1

λ) + (λR)2+σ + ‖φ‖p∗

). (24)

By Lemma 2.1, (19) and (24), we have

‖A(φ)‖∗ ≤ C‖N(φ)‖∗∗ + C‖K(x

λ)(W + φ)p‖∗∗

≤ C(‖φ‖2∗ + ‖φ‖p∗ + a(R) + a(

1

λ) + (λR)2+σ

)≤ C

(ρ2 + ρp + a(R) + a(

1

λ) + (λR)2+σ

).

(25)

Now we choose ρ small enough, such that C(ρ2 + ρp) ≤ 14ρ. Then choose R large

enough such that Ca(R) ≤ 14ρ. Finally we choose λ small enough such that

Ca( 1λ ) + C(λR)2+σ ≤ 1

2ρ. All yield that A(F) ⊂ F.It remains to prove that A is contractible.Similar to arguments in [4], we see that ∀φ1, φ2 ∈ F,

‖N(φ1)−N(φ2)‖∗∗ ≤ C(ρ+ ρp−1)‖φ1 − φ2‖∗. (26)

Observe that

|K(x

λ)||(W + φ1)p − (W + φ2)p|

≤C|K(x

λ)||φ1 − φ2|(|W |p−1 + |φ1|p−1 + |φ2|p−1).

Similarly we obtain

sup|x|≤λR

|x|2+σ|K(x

λ)||(W + φ1)p − (W + φ2)p|

≤C‖φ1 − φ2‖∗ sup|x|≤λR

|x|2(|W |p−1 + |φ1|p−1 + |φ2|p−1)

≤C‖φ1 − φ2‖∗(

(λR)2 + ρp−1),

(27)

supλR≤|x|≤1

|x|2+σ|K(x

λ)||(W + φ1)p − (W + φ2)p| ≤ Ca(R)‖φ1 − φ2‖∗, (28)

sup|x|≥1

|x|2+2p−1 |K(

x

λ)||(W + φ1)p − (W + φ2)p| ≤ Ca(

1

λ)‖φ1 − φ2‖∗. (29)

Hence by Lemma 2.1, (26)–(29), we have

‖A(φ1)−A(φ2)‖∗

≤C(

(‖N(φ1)−N(φ2)‖∗∗ + ‖K(x

λ)[(W + φ1)p − (W + φ2)p]‖∗∗

)≤C‖φ1 − φ2‖∗

(ρ+ ρp−1 + (λR)2 + a(R) + a(

1

λ))

≤1

2‖φ1 − φ2‖∗,

(30)

provided that ρ small enough, R large enough and λ small enough.By (25) and (30), A is a contraction mapping. By contraction-mapping principle,

it follows that A has a fixed point φλ in F. Hence W + φλ is a solution of∆u+ up + K(

x

λ)up = 0,

u > 0 in RN , lim|x|→∞

u(x) = 0.(31)

INFINITE MULTIPLICITY 1249

For x such that |x| = 1,W + φλ remains bounded because φλ(x) ≤ C. Thenuniform upper bound for W + φλ follows from (31) by observing that‖(1 + K(xλ ))(W + φλ)p‖Lq(B1) remains bounded as λ→ 0 for q > N

2 . In fact,∫B1

(1 + K(x

λ))q(W + φλ)pq ≤ C

∫B1

W pq + C

∫B1

|φλ|pq

≤ C + C

∫B1

|x|−σpq ≤ C

provided that σ > 0 small. Hence

|W + φλ| ≤ C for all |x| ≤ 1. (32)

It follows then that

|φλ(x)| ≤ C for all x. (33)

Thus uλ(x) = K− 1p−1

∞ λ2p−1 (W (λx) + φλ(λx)) is a continuum solutions of (1) and

limλ→0

uλ(x) = 0

uniformly in RN . This ends the proof of Theorem 1.1.

Remark 2. We observe that the above proof actually applies with no changes to thecase N+2

N−2 < p < N+1N−3 provided that K is symmetric with respect to N coordinate

axis, namely

K(x1, . . . , xi, . . . , xN ) = K(x1, . . . ,−xi, . . . , xN ), for all i = 1, . . . , N.

In this case the problem is invariant with respect to the above reflections, and wecan formulate the fixed point problem in the space of functions with these evensymmetries with the linear operator defined in Lemma 2.2. Indeed, the orthogonalityconditions in Lemma 2.2 are automatically satisfied, so that the associated numbersci’s are all zero.

4. The proof of Theorem 1.2. In this section, we consider the case when p ∈(N+2N−2 ,

N+1N−3 ] and prove Theorem 1.2. We need to use Lemma 2.2 and a Liapunov-

Schmidt reduction argument.By Lemma 2.2 and Remark 1, there is an obstruction in the solvability of the

linearized operator. To overcome the obstruction, we introduce a new parame-ter ξ to be determined later. For this reason we make the change of variables

K1p−1∞ λ−

2p−1u(x−ξλ ) and look for a solution of the form u = W + φ, leading to the

following equation for φ:

∆φ+ pW p−1φ = N(φ)− K(x− ξλ

)(W + φ)p, (34)

where

N(φ) = −(W + φ)p +W p + pW p−1φ.

We will change slightly the previous notations to make the dependence of the normson σ explicit. Hence we set

‖φ‖(σ)∗,ξ = sup|x−ξ|≤1

|x− ξ|σ|φ(x)|+ sup|x−ξ|≥1

|x− ξ|2p−1 |φ(x)|

and

‖h‖(σ)∗∗,ξ = sup|x−ξ|≤1

|x− ξ|σ+2|h(x)|+ sup|x−ξ|≥1

|x− ξ|2p−1+2|h(x)|.

1250 LIPING WANG AND JUNCHENG WEI

In the rest of the section we assume that

N + 2

N − 2< p <

N + 1

N − 3.

The case p = N+1N−3 can be handled similarly, with a slight modification of the norms

where it is more convenient to define

‖φ‖(σ)∗,ξ = sup|x−ξ|≤1

|x− ξ|σ|φ(x)|+ sup|x−ξ|≥1

|x− ξ|2p−1+α|φ(x)|

and

‖h‖(σ)∗∗,ξ = sup|x−ξ|≤1

|x− ξ|σ+2|h(x)|+ sup|x−ξ|≥1

|x− ξ|2p−1+α+2|h(x)|

for some small fixed α > 0, see Remark 1 and Remark 3.The proof of Theorem 1.2 is through a Liapunov-Schmidt reduction procedure.

This will be achieved in two steps. In the first step, we solve (34) modulo Zi,using Lemma 2.2. That is, we have the following lemma.

Lemma 4.1. Assume that N ≥ 4, N+2N−2 < p < N+1

N−3 and p > N+2N−2 if N = 3, K(x)

satisfies (H) and λ > 0. Then there is λ0 > 0 such that for |ξ| ≤ Λ and λ < λ0there exist φλ, c1(λ), . . . , cN (λ) solution to

∆φ+ pW p−1φ = N(φ)− K(x− ξλ

)(W + φ)p +

N∑i=1

ciZi,

lim|x|→∞

φ(x) = 0.

(35)

If K(x) also satisfies

|K(x)−K∞| ≤ C|x|−µ, |x| ≥ 1, (36)

for some µ > 0, then for 0 < θ < N − 2,

‖φλ‖(θ)∗,ξ +

N∑i=1

|ci(λ)| ≤ Cλmin{µ,2+θ} for all 0 < λ < λ0. (37)

Proof. Similar to the proof of Theorem 1.1, we fix 0 < σ < min {2, 2p−1} and define

for small ρ > 0

F = {φ : RN → R | ‖φ‖(σ)∗,ξ ≤ ρ}and the operator φ1 = Aλ(φ) to

∆φ1 + pW p−1φ1 = N(φ)− K(x− ξλ

)(W + φ)p +

N∑i=1

ciZi,

lim|x|→∞

φ1(x) = 0.

(38)

By the same proof as in those of Theorem 1.1, we have for any φ, φ1, φ2 ∈ F

‖N(φ)‖(σ)∗∗,ξ ≤ C(‖φ‖(σ)∗,ξ )2 + C(‖φ‖(σ)∗,ξ )p ≤ C(ρ2 + ρp), (39)

‖N(φ1)−N(φ2)‖(σ)∗∗,ξ ≤ C(ρ+ ρp−1)‖φ1 − φ2‖(σ)∗,ξ , (40)

‖K(x− ξλ

)(W + φ)p‖(σ)∗∗,ξ ≤ C(a(R) + a(

1

λ) + (λR)2+σ + (‖φ‖(σ)∗,ξ )p

), (41)

INFINITE MULTIPLICITY 1251

and

‖K(x− ξλ

)(

(W + φ1)p − (W + φ2)p)‖(σ)∗∗,ξ

≤C(a(R) + a(

1

λ) + (λR)2 + ρp−1

)‖φ1 − φ2‖(σ)∗,ξ .

(42)

Using Lemma 2.2 and fixed point theorem we get a solution φλ, c1(λ), . . . , cN (λ) of(35) provided ρ small enough, R large enough and λ small enough.

As in the proof of (33), we can obtain

|φλ| ≤ C for all x. (43)

Under the assumption of (36) and for 0 < θ < N − 2, we can estimate

K(x−ξλ )(W + φλ)p as follows: for R fixed large enough,

sup|x−ξ|≤1

|x− ξ|2+θ|K(x− ξλ

)(W + φλ)p| ≤ sup|x−ξ|≤λR

· · ·+ supλR≤|x−ξ|≤1

· · ·

sup|x−ξ|≤λR

|x− ξ|2+θ|K(x− ξλ

)(W + φλ)p|

≤C sup|x−ξ|≤λR

|x− ξ|2+θ(|W |p + |φλ|p)

≤C(

(λR)2+θ + ‖φλ‖(θ)∗,ξ sup|x−ξ|≤λR

|x− ξ|2)

≤C(

(λR)2+θ + ‖φλ‖(θ)∗,ξ(λR)2),

(44)

supλR≤|x−ξ|≤1

|x− ξ|2+θ|K(x− ξλ

)(W + φλ)p|

≤Cλµ supλR≤|x−ξ|≤1

|x− ξ|2+θ−µ(|W |p + |φλ|p)

≤Cλmin{µ,2+θ} + Cλmin{µ,2}‖φλ‖(θ)∗,ξ,

(45)

sup|x−ξ|≥1

|x− ξ|2+2p−1 |K(

x− ξλ

)(W + φλ)p|

≤Cλµ(

1 + (‖φλ‖(θ)∗,ξ)p)

sup|x−ξ|≥1

|x− ξ|−µ

≤Cλµ(

1 + (‖φλ‖(θ)∗,ξ)p).

(46)

Thus

‖K(x− ξλ

)(W + φλ)p‖(θ)∗∗,ξ ≤ Cλmin{µ,2+θ} + Cλmin{µ,2}‖φλ‖(θ)∗,ξ. (47)

If 0 < θ ≤ 2p−1 , according to (19) and (47), Lemma 2.2 yields

‖φλ‖(θ)∗,ξ +

N∑i=1

|ci(λ)| ≤ C(‖N(φλ)‖(θ)∗∗,ξ + ‖K(

x− ξλ

)(W + φλ)p‖(θ)∗∗,ξ

)≤C

((‖φλ‖(θ)∗,ξ)

2 + (‖φλ‖(θ)∗,ξ)p + λmin{µ,2+θ} + λmin{µ,2}‖φλ‖(θ)∗,ξ

)which leads to

‖φλ‖(θ)∗,ξ +

N∑i=1

|ci(λ)| ≤ Cλmin{µ,2+θ}. (48)

1252 LIPING WANG AND JUNCHENG WEI

provided ρ, λ small enough.Now consider 2

p−1 < θ < N − 2 and let 0 < σ ≤ 2p−1 .

If p ≥ 2, then 0 < σ ≤ 2 and

|N(φλ)| ≤ C(W p−2|φλ|2 + |φλ|p).

Observe that

sup|x−ξ|≤1

|x− ξ|2+θ|N(φλ)| ≤ sup|x−ξ|≤λ

· · ·+ supλ≤|x−ξ|≤1

· · ·

Thanks to (43), we have

sup|x−ξ|≤λ

|x− ξ|2+θ|N(φλ)| ≤ Cλ2+θ

and

supλ≤|x−ξ|≤1

|x− ξ|2+θ|N(φλ)| ≤C(‖φλ‖(σ)∗,ξ )2 supλ≤|x−ξ|≤1

|x− ξ|2+θ−2σ

≤C(‖φλ‖(σ)∗,ξ )2 supλ≤|x−ξ|≤1

|x− ξ|2−σ

≤C(‖φλ‖(σ)∗,ξ )2 ≤ Cλmin{2(2+σ),2µ},

sup|x−ξ|≥1

|x− ξ|2+2p−1 |N(φλ)| ≤ C(‖φλ‖(σ)∗,ξ )2 ≤ Cλmin{2(2+σ),2µ}.

Thus

‖N(φλ)‖(θ)∗∗,ξ ≤ Cλmin{2+θ,2(2+σ),2µ} if p ≥ 2. (49)

Similarly, if 1 < p < 2, using |N(φλ)| ≤ C|φλ|p, we can get

‖N(φλ)‖(θ)∗∗,ξ ≤ Cλmin{2+θ,p(2+σ),pµ}. (50)

After finite steps we get for any p > 1,

‖N(φλ)‖(θ)∗∗,ξ ≤ Cλmin{2+θ,µ}. (51)

According to (47),(51) we get

‖φλ‖(θ)∗,ξ +

N∑i=1

|ci(λ)| ≤ Cλmin{µ,2+θ}. (52)

provided λ small.

In the second step, we need to vary ξ so that ci = 0, i = 1, ..., N , therefore provingTheorem 1.2.

By Lemma 4.1, we have found a solution φλ, c1(λ), . . . , cN (λ) to (35). By Lemma2.2 the solution constructed satisfies for all 1 ≤ j ≤ N :∫

RN

(N(φλ)− K(

x− ξλ

)(W + φλ)p)∂W∂xj

= 0

if and only if cj = 0. For this lots of wonderful analysis is involved. We divide itinto three cases.Case (a): µ > N. In this case, we have as λ→ 0∫

RN−K(

x

λ)W p ∂W

∂xj(x+ ξ) = − λN

K∞

∫RN

(K(x)−K∞

)W p(ξ)

∂W

∂xj(ξ) + o(λN ),

where the convergence is uniform with respect to |ξ| ≤ δ0.

INFINITE MULTIPLICITY 1253

Indeed, in the case p ≥ 2, if we choose N−22 < θ < min{N2 , N − 2}, then we

obtain ∫RN

∣∣∣∣N(φλ)∂W

∂xj

∣∣∣∣ =

∫B1(ξ)

· · ·+∫RN\B1(ξ)

· · ·

∫B1(ξ)

∣∣∣∣N(φλ)∂W

∂xj

∣∣∣∣ ≤ C(‖φλ‖(θ)∗,ξ)2

∫B1(ξ)

|x− ξ|−2θ

≤ Cλ2min{2+θ,µ} ≤ Cλmin{N+2,2µ},∫RN\B1(ξ)

∣∣∣∣N(φλ)∂W

∂xj

∣∣∣∣ ≤ C(‖φλ‖(θ)∗,ξ)2

∫RN\B1(ξ)

|x− ξ|−3−4p−1

≤ Cλ2min{2+θ,µ} ≤ Cλmin{N+2,2µ}.

Thus ∫RN

∣∣∣∣N(φλ)∂W

∂xj

∣∣∣∣ = o(λN ). (53)

Similarly, in the case 1 < p < 2, we get∫RN

∣∣∣∣N(φλ)∂W

∂xj

∣∣∣∣ ≤ Cλpmin{2+θ,µ} = o(λN ), (54)

since we can choose θ such that pθ = N−1, which is possible since then θ = N−1p <

(N−1)(N−2)N+2 < N − 2.

Next we need the following important lemma

Lemma 4.2. 0 < c ≤ Uλ ≤ C in B1(ξ), where Uλ = W + φλ.

Proof. The upper bound has been given by (43) and we only need to prove that Uλhas a lower bound. Let χ(r) = 1

2N (1− r2), so that

∆χ ≡ −1, χ ≡ 0 on ∂B1

and consider z = Uλ + (∑Ni=1 |ci(λ)|‖Zi‖∞)χ. Then z satisfies

∆z ≤ 0.

By maximum principle we have

Uλ + (

N∑i=1

|ci(λ)|‖Zi‖∞)χ ≥ Uλ |∂B1≥ W (1)

2in B1,

since the convergence φλ → 0 as λ→ 0 is uniform on any compact set of RN\{0}.Then Uλ ≥ W (1)

4 > 0 in B1 since ci(λ) → 0 as λ → 0 and χ(r), Zi(1 ≤ i ≤ N) arebounded. Thus we proved this lemma.

Define now Fλ(ξ) by

F(j)λ (ξ) :=

∫RN−K(

x

λ)Upλ

∂W

∂xj(x+ ξ) +

∫RN

N(φλ)∂W

∂xj

∼− λN∫RN

(K(x)−K∞

)∂W∂xj

(ξ) + o(λN )

(55)

and Fλ(ξ) =(F

(1)λ (ξ), . . . , F

(N)λ (ξ)

). Fix now δ > 0 small. Then from (53)–(55),

we have for small λ〈Fλ(ξ), ξ〉 6= 0 for all |ξ| = δ.

1254 LIPING WANG AND JUNCHENG WEI

By degree theory we deduce that Fλ(ξ) has a zero point in Bδ(0).

Case (b.1): N − 2p+2p−1 < µ < N .

Obviously∫RN

K(x− ξλ

)W p ∂W

∂xj∼ λµ

∫RN

f(x

|x|)|x|−µW p(x+ ξ)

∂W

∂xj(x+ ξ).

By the above computations, for θ1, θ2 ∈ (0, N − 2), we have∫RN

∣∣∣∣N(φλ)∂W

∂xj

∣∣∣∣ = O(λ2min{2+θ1,µ} + λpmin{2+θ2,µ}

).

If we choose 2θ1 = µ− 2 and pθ2 = µ which are possible since µ < N and p > N+2N−2 ,

then ∫RN

∣∣∣∣N(φλ)∂W

∂xj

∣∣∣∣ = o(λµ), (56)

On the other hand ∫RN

∣∣∣∣K(x− ξλ

)((W + φλ)p −W p

)∂W∂xj

∣∣∣∣=

∫RN\B1(ξ)

· · ·+∫B1(ξ)\BλR(ξ)

· · ·+∫BλR(ξ)

· · ·

∫BλR(ξ)

∣∣∣∣K(x− ξλ

)((W + φλ)p −W p

)∂W∂xj

∣∣∣∣≤C

∫BλR(ξ)

(|φλ|p + |φλ|) ≤ C‖φλ‖(θ)∗,ξ∫BλR(ξ)

|x− ξ|−θ

≤C‖φλ‖(θ)∗,ξ(λR)N−θ ≤ Cλmin{2+θ,µ}+N−θ = o(λµ),

(57)

∫B1(ξ)\BλR(ξ)

∣∣∣∣K(x− ξλ

)((W + φλ)p −W p

)∂W∂xj

∣∣∣∣≤Cλµ

∫B1(ξ)\BλR(ξ)

|x− ξ|−µ(|φλ|p + |φλ|)

≤Cλµ‖φλ‖(θ)∗,ξ∫B1(ξ)\BλR(ξ)

|x− ξ|−µ−θ

≤Cλµ‖φλ‖(θ)∗,ξ(λR)N−µ−θ ≤ Cλµ+min{2+θ,µ}+N−µ−θ = o(λµ),

(58)

∫RN\B1(ξ)

|K(x− ξλ

)((W + φλ)p −W p

)∂W∂xj|

≤Cλµ‖φλ‖(θ)∗,ξ∫RN\B1(ξ)

|x− ξ|−3−4p−1−µ

≤Cλµ+min{2+θ,µ} = o(λµ).

(59)

INFINITE MULTIPLICITY 1255

Thus ∫RN

(N(φλ)− K(

x− ξλ

)(W + φλ)p)∂W∂xj

=−∫RN

K(x− ξλ

)W p ∂W

∂xj+ o(λµ)

∼− λµ∫RN|x|−µf(

x

|x|)W p(x+ ξ)

∂W (x+ ξ)

∂xj+ o(λµ).

(60)

Define now F to be given by

F (ξ) := − 1

p+ 1

∫RN|x|−µf(

x

|x|)W (x+ ξ)p+1.

By Dominate Convergence Theorem, as |ξ| → ∞ we get

F (ξ) = −βp+1p−1

p+ 1|ξ|N−µ−

2(p+1)p−1

∫RN|x|−µf(

x

|x|)|x+

ξ

|ξ||−

2(p+1)p−1 + o

(|ξ|N−µ−

2(p+1)p−1

)and

∇F (ξ) · ξ

=− βp+1p−1

p+ 1(N − µ− 2(p+ 1)

p− 1)|ξ|N−µ−

2(p+1)p−1

∫RN|x|−µf(

x

|x|)|x+

ξ

|ξ||−

2(p+1)p−1

+ o(|ξ|N−µ−

2(p+1)p−1

).

Therefore ∇F (ξ) · ξ 6= 0 for all |ξ| = R where R large. Using this and degree theorywe obtain the existence of ξ such that cj = 0, 1 ≤ j ≤ N provided λ small enough.

Case (b.2): µ = N .In this case, we will have for j = 1, . . . , N,

Gj(ξ) :=

∫RN−K(

x

λ)Upλ

∂W

∂xj(x+ ξ) +

∫RN

N(φλ)∂W

∂xj

=

∫RN−K(

x

λ)Upλ

∂W

∂xj(x+ ξ) + o(λN )

(61)

uniformly for ξ on any compact subset of RN .Similar to case (a), we derive that for small fixed ρ

〈G(ξ), ξ〉 6= 0 for all |ξ| = ρ, (62)

where G(ξ) = (G1(ξ), . . . , GN (ξ)).Indeed, for ρ > 0 small it holds

〈∇W (ξ), ξ〉 < 0 for all |ξ| = ρ.

Thus, for δ > 0 small and fixed

γ ≡ supx∈Bδ(0)

〈∇W (x+ ξ), ξ〉 < 0 for all |ξ| = ρ. (63)

We decompose∫RN−K(

x

λ)Upλ〈∇W (x+ ξ), ξ〉 =

∫Bδ

· · ·+∫RN\Bδ

· · ·

1256 LIPING WANG AND JUNCHENG WEI

where∣∣∣∣∣∫RN\Bδ

−K(x

λ)Upλ〈∇W (x+ ξ), ξ〉

∣∣∣∣∣ ≤ CλN∫|x|≥δ

|x|−N |x|−2p−1−1 ≤ CλN . (64)

On the other hand, for R > 0 we may write∫Bδ

−K(x

λ)Upλ〈∇W (x+ ξ), ξ〉 =

∫Bδ\BλR

· · ·+∫BλR

· · ·

We have ∫BλR

−K(x

λ)Upλ〈∇W (x+ ξ), ξ〉 = O(λN ). (65)

Since, by Lemma 4.2, we can get that∫Bδ\BλR

−K(x

λ)Upλ〈∇W (x+ ξ), ξ〉 ∼

∫Bδ\BλR

K(x

λ). (66)

But ∫Bδ\BλR

K(x

λ) =λN

∫Bδ\BλR

|x|−Nf(x

|x|)

+ λN∫Bδ\BλR

|x|−N(λ−N |x|NK(

x

λ)− f(

x

|x|)) (67)

and

λN∫Bδ\BλR

|x|−Nf(x

|x|) = λN log

1

λ

∫SN−1

f +O(λN ) (68)

while given any ε > 0 there is R > 0 such that∣∣∣∣∣λN∫Bδ\BλR

|x|−N(λ−N |x|NK(

x

λ)− f(

x

|x|))∣∣∣∣∣ ≤ ελN log

1

λ. (69)

From (64)–(69) we deduce the validity of (62). Applying again degree theory weconclude that for some |ξ| < ρ we have G(ξ) = 0.

Remark 3. The proof of Theorem 1.2 in the case p = N+1N−3 follows exactly the same

lines with the modified norms as defined in Remark 1. The argument works becausewe assume that K(x)−K∞ has decay, which implies that even the modified norms,

the error ‖K(W + φ)p‖(σ)∗∗,ξ converges to 0. Indeed, we have

sup|x|≥1

|x− ξ|2+2p−1+α|K(

x− ξλ

)(W + φ)p| ≤ Cλµ sup|x|≥1

|x− ξ|α−µ = Cλµ

provided that α < µ.

Acknowledgments. The research of the first author is partially supported byNSFC 10901053 and the second author is partially supported by an EarmarkedGrant from RGC of Hong Kong. We thank Professor W.-M. Ni for useful discussionsand also thank the referee for making good suggestions.

INFINITE MULTIPLICITY 1257

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Received December 2011; revised July 2012.

E-mail address: lpwang@math.ecnu.edu.cn

E-mail address: wei@math.cuhk.edu.hk