Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/89/020245-13 5; 1.50 + 0.20/0 Vol. 40, March 1989 �9 1989 Birkh/iuser Verlag, Basel

Semilinear elliptic problems in annular domains

By C. Bandle, Mathematics Institute, University of Basel, Basel, CH-4051, Switzerland, and Man Kam Kwong*?, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4844, USA

I. Introduction

In this paper, we extend the study of semilinear elliptic boundary value problems in annular domains undertaken by Bandle, Coffman and Marcus in [1]. They are interested in the question of existence and uniqueness of positive radially symmetric (p.r.s.) solutions of the reaction-diffusion equa- tion

Au +f(u) = 0 in R, < Ix I < Ro, x~-R N, N -> 3 (1.1)

subject to one of the following sets of boundary conditions:

u = 0 on [x[=Rl and [x[=Ro, (1.2a)

u = 0 on [x[=R, and ~ r = 0 on Ix[=Ro, (1.2b)

or=O on [xl=R, and u=O on Ixl=Ro. (1.2c)

Here r = Ixi and gfi?r denotes differentiation in the radial direction. The problems under consideration reduce to problems in ordinary

differential equations. With a suitable change of the independent variable, (1.1) becomes an equation of the Emden-Fowler type:

2N - 2 u"(t)+t~f(u)=O t o < t < t , , c~= N - 2 (1.3)

* Permanent address: Dept. of Mathematical Sciences, Northern Illinois University, DeKalb, II, 60115-2888, USA. t This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38. The author also wants to thank the Mathematics Institute, University of Basel, for supporting a visit in Oct., 1987 during which the project was initiated.

246 C. Bandle and M. K. Kwong

and the boundary conditions (1.2a)-(1.2c) become

ZAMP

U(to) = u(tO = 0, (1.4a)

u'(to) = u(fi) = O, (1.4b)

U(to) = u ' ( t , ) = O, (1.4c)

respectively. In this paper, we consider, instead of (1.3), the more general equation

u"(t) + F( t , u) = 0 (1.5)

in which F(t , u) is sufficiently smooth. In [1], it is proved that problem (1.1)-(1.2a) (respectively (1.2b), (1.2c))

always possesses a p.r.s, solution if the following three conditions are satisfied:

(A-I) f is nondecreasing in (0, 0o).

(A-2) lim f ( u ) / u = 0o.

(A-3) l im f ( u ) / u = O. u-~0

It is remarked in [1] that (A-l) is not a necessary condition for existence. This has been confirmed by Coffman and Marcus [4] who showed that (A-2) and (A-3) are sufficient to guarantee existence. We give here a different proof which doesn't require the positivity of f and applies as the one of Coffman and Marcus to the more general equation (1.5).

Another result established in [1] is that assuming condition (A-3) alone, the existence of a p.r.s, solution for problem (1.1), (1.2a) in every annulus is equivalent to the existence of a p.r.s, solution for boundary value problem (1.1), (1.2b) in every annulus, and implies existence for boundary value problem (1.1), (1.2c). We shall see that in fact all three existence assertions are equivalent.

As in [1], the approach used is the shooting method. However, instead of shooting the solution from an endpoint where a Dirichlet condition is imposed, we shoot either from an endpoint where a Neumann condition is satisfied or from an interior point with a Neumann condition.

Uniqueness of the p.r.s, solution of the various problems does not hold in general. Most known uniqueness criteria apply only to superlinear equations. These are equations in which the nonlinear term F(t , u) satisfies

(S) F(t , u)/u > 0 and is nondecreasing in (0, 0o) for each fixed t, and nonconstant in any interval (0, e), e > 0.

This condition is necessary for the Kolodner-Coffman method to work.

Vot. 40, 1989 Semilinear elliptic problems in annuli 247

For a survey of this useful method, see [6]. In the special case (1.1), superlinearity is implied by the stronger condition

3~ > 0 such that f (u) /u 1+~ is monotone increasing in (0, oo) (1.6)

It is proved in [1] that under this assumption, uniqueness in every annulus for problem (1.1), (1.2a) implies uniqueness for problem (1.1), (1.2c). In Section 4 we show that the result holds for a much wider class of equations under the weaker condition (S) instead of (1.6).

In a recent paper of Coffman and Marcus [4] uniqueness is proved for (1.1), (1.2a) under the assumptions (A-2), (A-3) and (S). In a recent paper, Garaizar [5] treated the existence problem with the phase-plane method.

2. Equivalence of existence for the problems

In this section, we shall show that for a wide class of equations, under condition (F-3), the existence of a p.r.s, solution for the three boundary value problems are equivalent.

For the ease of referencing we repeat below the boundary value prob- lems that we are interested in

u " + F ( t , u ) = O , u > 0 O < t o < t < t l (2.1)

U(to) = u(tl) = 0, (2.2a)

u'(to) = u(fi) = 0, (2.2b)

U(to) = u'(tl) = 0. (2.2c)

We would refer to these problems as (BVP1,) (BVP2), and (BVP3) respectively. Various conditions will be imposed on the reaction term F(t, u). We mention three of them now.

(F-l) F is C O in its first variable and C ~ in its second.

(F-2) lira F(t, u)/u = oe uniformly on every [to, tl]. u --~ o o

(F-3) lim F(t, u)/u < 0 uniformly on every [to, tl]. u ~ 0

Throughout this section, we assume conditions (F-l) and (F-3), but not (F-2). Notice that (F-2) corresponds to (A-2) in Section 1, (F-3) is weaker than (A-3), while (A-l) has no counterpart here. In particular, the function F(t, u) is not required to be positive for all values of t and u. If (F-2) is satisfied, then F(t, u) is positive for large values of u, but is allowed to assume negative values when u is small. Implicit in (F-3) is the requirement that the limit involved exists and is finite. Uniform convergence of limu~o F(t, u)/u dictates that F(t, u)/u is uniformly bounded from below.

248 C. Bandle and M. K. Kwong ZAMP

Uniform convergence in (F-2) and (F-3) are imposed only for technical reasons. The proofs given in this paper can be refined to handle the more general si tuation in which only pointwise convergence holds and that F(t, u)/u is uniformly bounded f rom below, and, for finite u, f rom above. We omit the details.

Let us first examine problem (BVP2). According to the method of shooting, we consider instead of a boundary value problem, an associated initial value problem. More precisely, we use to as our initial point, and find a solution u~(t) of (2.1) satisfying the initial condit ions

U(to) = a, u'(to) = 0. (2.3)

F r o m the classical theory of initial value problems, such a solution exists, is unique and depends cont inuously on the parameter c~, the initial height of the solution. We may also assume wi thout loss of generality that u~(t) has been extended to a maximal domain of definition.

It is easy to see that the uniform boundedness f rom below of the ratio F(t, u)/u (by (F-3)) implies that u~(t) is uniformly bounded from above in [to, fi]. It may happen that u~(t) --, - ~ at some point t = t before reaching the left endpoint tl, since no further restriction on F is imposed when u takes on negative values. I f this does not happen, then u~ is defined th roughout the interval [to, td. In any case, the solution must intersect either the line segment X = [to, t,] x {0} on the t-axis, or the upper ray Y = {t~} • [0, ~ ) of the straight line t = tl. Let us denote by L the union of the line segment X and the ray Y, and by T(a) = (Tl(a), Tz(a)) the first point of intersection of u~ with L. Figures 1 and 2 show two ways the intersection can occur. I f T(a) happens to be the point (t~, 0), then u~ is a solution of our original boundary value problem (BVP2).

Lemma 1. T(a) is a cont inuous function of ~.

Proof. I f T(a) lies on Y, the conclusion is simply a consequence of the cont inuous dependence of u~(tl) on ~. I f T(a) lies on X, the conclusion is a

t o T ( a ) t 1

Figure 1. Figure 2.

T(a)

Vol. 40, 1989 Semilinear elliptic problems in annuli 249

consequence of the inverse function theorem provided we can show that u~ does not touch the t-axis tangentially. The latter assertion follows from the uniform boundedness from below of F(t, u)/u. []

A simple continuity argument is what makes the shooting method work.

Lemma 2. If there exist two initial heights ~1 and e2 such that T(0~l) lies on Y and T(,2) lies on X, then the boundary value problem (2.1), (2.2b) has a solution.

Let us show next that (F-3) implies the existence of ~ as needed in Lemma 2.

Lemma 3. Suppose that (F-3) holds. Then

u~(t) <- c~ cosh(k( t l - to)), te[t0, Tl(e)],

where - k 2 is a lower bound of F(t, u)/u on [to, t~].

(2.4)

Proof. The solution u~ is non-negative in the interval [to, T~(e)]. The conclusion is a straight-forward estimate using the inequality

u " - k 2 u <- O, te[t0, T~(e)]. [] (2.5)

By choosing the initial height e small enough, we can make sure that the righthand side of (2.4) is so small that by (F-3)

f ( t , u) <- 4(t, - to) u, te[to, T,(e)l. (2.6)

Lemma 4. For all sufficiently small c~ such that (2.6) holds, T(e) lies on Y.

Proof. In the interval [to, T~(c0], (2.6) yields the differential inequalities 2

u " + 4(tl to) u (2.7)

This can be compared

I r c ]2z (2.8) z" + 4( t l - t o )

with initial conditions

z'(to) = O, Z(to) = cc (2.9)

We know from the classical Sturm comparison theorem that u~ oscillates less than z. Since z has no zero before t~, u cannot have a zero before t~. Thus T(~) must lie on Y. []

->0.

with the differential equation

= 0,

250 C. Bandle and M. K. Kwong ZAMP

In the next section, we shall show that (F-2) implies the existence of an ~2 having the property in Lemma 2. It then follows that (F-1)-(F-3) imply existence of a solution of the boundary value problem in question.

The following corollary is an easy consequence of Lemmas 1, 2 and 4.

Corollary 1. If (BVP2) for an interval [to, t~] has a solution, then (BVP2) for any larger interval [to, t2], t2 > t~ also has a solution.

In [1], the shooting method is also used, but the initial point is chosen to be t~ (the solution is shot backwards towards to) and the solution u~(t) is given the initial slope - f l :

ut3(tl) = O, u'~(t~) = - f t . (2.10)

As before, we can show that the solution can be extended to a maximal domain of definition, and that it must first intersect /S, the union of the line segment X and the upper ray Y = {to} • [0, ~ ) at some point T(fl) = (T~(fl), Tz(fl)). The slope of the solution S(fl) = u'~(T~(fl)) at the point of intersection is a continuous function of the parameter ft.

Lemma 5. If there exist two initial slopes - f l l and --f12, such that T(fl~) lies on Y and S(fl~) < 0, and either ]r(fl2) lies on X or S(fl2) > 0, then problem (2.1), (2.2b) has a solution.

On the other hand, if there exist two initial slopes -fl~ and -f12, such that it(ill) lies on Y and ]r(fl2) lies on X, then problem (2.1), (2.2a) has a solution.

Proof. We prove only the first half of the lemma. Notice that if T(fl2) lies on X, S(fl2) must be positive; it is non-negative because u B is positive for t > ]r~(fl2), and it cannot be zero according to the uniqueness theorem of ordinary differential equations. A simple continuity argument shows that as fl varies from fll to f12, there exists a value for which S ( f l ) = 0. As seen above, the point of intersection ]r(fl) cannot lie on the t-axis. The function uB(t ) is thus a solution of the boundary value problem. �9

In the study of (BVP1) or (BVP3), shooting can be started at the left endpoint and proceeds similarly.

An estimate for u B on [Tl(fi), tl] analogous to (2.4) can be easily obtained using s inh (k (q - to)). With this estimate, we can establish the analog of Lemma 4.

Lelnma 6. For sufficiently small fl, ]r(fl) lies on I 7.

Corollary 2. If (BVP1) has a solution in any interval [to, tl], it has a solution in any larger interval.

Vol. 40, 1989 Semilinear elliptic problems in annuli 251

Our first theorem states that under no further condit ions on F(t, u), existence of a solution for (BVP1) in all annulus implies existence of a solution for the other two boundary value problems.

Theorem 1. Suppose that F(t, u) satisfies condit ions (F- I ) and (F-3). If it is known that the boundary value problem (2.1), (2.2a) always has a solution for all given values of to and t~, then the boundary value problem (2.1), (2.2b) (respectively (2.2c)) always has a solution.

Proof. We give only the p roo f concerning (BVP2). Choose any t2 < to. By hypotheses, (BVP1) for the intervals [t2, to] and [t2, t~] each has a solution (denoted by U(t) and 0 ( 0 respectively). Regard these two solutions as being shot out at t2 with initial slopes ?t and 72 respectively. As the initial slope varies f rom 7~ to 72, the end point of the solution moves f rom to towards t~. By changing 7~ and 72 if necessary, we can assume without loss of generality that this endpoint always lies in [to, fi]. Obviously U'(to) < 0. By choosing 7 between 7l and 72 and close enough to 7~, we can make sure that the solution x(t) shot out f rom t2 lands at a point t3e(to, tl) and that x'(to) < 0. Regard x(t) = u~ as the solution being shot out at t3 with initial slope x'(t3) = -3~- Then S(/~l) = x'(to) < 0. Boundary value problem (BVP1) on [to, t3] has a solution, denoted by ua2(t), that is shot out at t3 with slope -f12. Since u~2(to ) = O, S(~2) = u'~2(to) > 0. By Lemm a 5, (BVP2) on [to, t3] has a solu- tion. Finally, by Corollary 1, (BVP2) on [to, t�91 has a solution, m

An extra condi t ion on F(t, u) is needed to establish the converse of Theorem 1.

(F-4) For any compact interval [to, t~] there exist a positive function ~b(u), a value u0 > 0 and positive constants C~, C2 such that Cld~(u) < F(t, u) < C2~(u) for all u > u0.

Examples of such functions are products , F(t, u )= q(t)f(u), where q(t) > 0 and f (u) > 0 for u > u0, and their sums.

Lemma 7. Suppose that (F-4) holds. Let u(t) be a monotonical ly decreasing solution of (BVP2), and v(t) be a solution of the initial value problem

v"(t) + qS(v) = 0, (2.11)

V(to) = U(to), v'(to) = 0. (2.12)

If u(z~) = Uo at a point z~e[t0, hi, then v(t) must decrease to the value u0 at a point ~ > to and

z - to < x/~2(r , - to). (2.13)

252 C. Bandle and M. K. Kwong ZAMP

Proof. By using a scaling in the independent variable t if necessary, we can assume without loss of generality that C2 = 1. Then equation (2.1) becomes the differential inequality

u"(t) + c~(u) > O. (2.14)

First note that by the positivity of 4~, v' < 0 in (to, v). By multiplying (2.11) by v' and (2.14) by u', we get after integration v ' Z / 2 - ~ ( u ( t o ) ) + �9 (v) = 0, ~ ' = q~, and in (to, z~) u'2/2 - ~(U(to)) + ~ (u ) < O. From this it follows that z < z~. So v must reach Uo before u does.

Lemma 8. Suppose that (F-4) holds. As in Lemma 7 let v(t) be the solution of (2.11) and it decreases to the value u0 at z. Let u(t) be a solution of (BVP3) o n [t2, to] and that (2.12) is satisfied. Then u(t) increases to Uo at a point zlE[t2, to] and

to -- Zl <- (z -- t o ) / x / ~ l . (2.15)

Proof. The proof is essentially identical to that of Lemma 7. Using scaling, we can assume without loss of generality that C~ = 1. It is known that v(t) must be monotonically decreasing in [to, ~]. The rest of the proof of Lemma 7 goes through with the roles of u and v exchanged. �9

Theorem 2. Under conditions (F-l) , (F-3) and (F-4), the boundary value problem (2.1), (2.2a) has a solution for all to and tl if and only if (2.1), (2.2b) has a solution for all to and t~ if and only if (2.1), (2.2c) has a solution for all to and t~.

Proof. In view of Theorem 1, we have only to prove that existence of a solution for (BVP2) on all intervals implies existence of a solution for (BVP1) on some given interval [to, fi]. By (F-4), F(t , u) > 0 for all t > 0 and u > u0. Choose a point s~( to , t~) so very close to t~ such that any solution shot out horizontally from t = s towards t = t~ at an initial height less than or equal to Uo + 1 will not have a zero in [s, t~]. This is possible because F(t , u) is uniformly bounded for t~[s , t~] and u~(0, u0 + 1]. We can further let s be so close to t~ that the point s~ = s - x/C22(t~ - s ) / x / / -~ is still close to t~ and thus far from to. By hypotheses, (BVP2) on [s, t,] has a solution u(t). Without loss of generality we may assume that u(t) is monotonically decreasing in Is, tl]. If it is not, we simply move s to the last local maximum of u(t) before t~. By the choice of s, we must have u(s) > u0 + 1. By Lemma 7, the solution v(t) of (2.11) with initial conditions v(s) = u(s) and v'(s) = 0 will decrease to the value Uo at a distance not too far from s. By applying Lemma 8 to compare u(t) to the left of t = s with v(t), we see that u(t) must decrease to the value u0 at some point a > s~. In the interval [a, s], the function u(t) is concave (since u " = - F ( t , u) < 0). Hence u'(a) > (u(s) - Uo)/ (s - a) > 1/(s - sO. By adjusting the choice of s we can guarantee that u'(a)

252 C. Bandle and M. K. Kwong ZAMP

Proof. By using a scaling in the independent variable t if necessary, we can assume without loss of generality that C2 = 1. Then equation (2.1) becomes the differential inequality

u"(t) + c~(u) > O. (2.14)

First note that by the positivity of 4~, v' < 0 in (to, v). By multiplying (2.11) by v' and (2.14) by u', we get after integration v ' Z / 2 - ~ ( u ( t o ) ) + �9 (v) = 0, ~ ' = q~, and in (to, z~) u'2/2 - ~(U(to)) + ~ (u ) < O. From this it follows that z < z~. So v must reach Uo before u does.

Lemma 8. Suppose that (F-4) holds. As in Lemma 7 let v(t) be the solution of (2.11) and it decreases to the value u0 at z. Let u(t) be a solution of (BVP3) o n [t2, to] and that (2.12) is satisfied. Then u(t) increases to Uo at a point zlE[t2, to] and

to -- Zl <- (z -- t o ) / x / ~ l . (2.15)

Proof. The proof is essentially identical to that of Lemma 7. Using scaling, we can assume without loss of generality that C~ = 1. It is known that v(t) must be monotonically decreasing in [to, ~]. The rest of the proof of Lemma 7 goes through with the roles of u and v exchanged. �9

Theorem 2. Under conditions (F-l) , (F-3) and (F-4), the boundary value problem (2.1), (2.2a) has a solution for all to and tl if and only if (2.1), (2.2b) has a solution for all to and t~ if and only if (2.1), (2.2c) has a solution for all to and t~.

Proof. In view of Theorem 1, we have only to prove that existence of a solution for (BVP2) on all intervals implies existence of a solution for (BVP1) on some given interval [to, fi]. By (F-4), F(t , u) > 0 for all t > 0 and u > u0. Choose a point s~( to , t~) so very close to t~ such that any solution shot out horizontally from t = s towards t = t~ at an initial height less than or equal to Uo + 1 will not have a zero in [s, t~]. This is possible because F(t , u) is uniformly bounded for t~[s , t~] and u~(0, u0 + 1]. We can further let s be so close to t~ that the point s~ = s - x/C22(t~ - s ) / x / / -~ is still close to t~ and thus far from to. By hypotheses, (BVP2) on [s, t,] has a solution u(t). Without loss of generality we may assume that u(t) is monotonically decreasing in Is, tl]. If it is not, we simply move s to the last local maximum of u(t) before t~. By the choice of s, we must have u(s) > u0 + 1. By Lemma 7, the solution v(t) of (2.11) with initial conditions v(s) = u(s) and v'(s) = 0 will decrease to the value Uo at a distance not too far from s. By applying Lemma 8 to compare u(t) to the left of t = s with v(t), we see that u(t) must decrease to the value u0 at some point a > s~. In the interval [a, s], the function u(t) is concave (since u " = - F ( t , u) < 0). Hence u'(a) > (u(s) - Uo)/ (s - a) > 1/(s - sO. By adjusting the choice of s we can guarantee that u'(a)

Vol. 40, 1989 Semilinear elliptic problems in annuli 253

is as large as we please. The uni form boundedness f rom below of the ratio F(t, u)/u enables us to conclude that if u'(a) is large enough, then when u(t) is cont inued beyond a towards to, it mus t intersect the t-axis at some point in [to, a]. Proofs of similar assertions can be found in the p roof of Theorem 3 in the next section. The function u(t) is thus a solution of (BVP1) in a subinterval of [to, t~]. By Corollary 2, (BVP1) for [to, t~] has a solution, i

3. Existence

In this section we assume that all three condit ions (F-1)- (F-3) hold. This is sufficient to guarantee that all three boundary value problems have solutions.

Theorem 3. Suppose (F-1)- (F-3) hold. Then all three boundary value problems (2.1), (2.2a/b/c) have solutions for any given to and tl.

Proof. We first look at (BVP2). As in section 2, we shoot the solution us(t) f rom t = to at an initial height ~. For convenience, we shall suppress the subscript ~ and write u(t) = u~(t). We claim that if ~ is sufficiently large, then T~(c0 lies between to and t~. Let 2l = tl - to.

F r o m condi t ion (F-2), there exists a value ul such that

f ( t , u) > u > ul, te[to, tl]. (3.1)

Let re(t0, t~) be the point where u (v )= ul. Within the interval [to, z], u(t) satisfies the differential inequality

u"(t) + 2l u <0. (3.2)

We can compare u(t) with the solution v(t) of

v ' ( t ) + ~ v = 0 , (3.3)

V(to) = ~, v'(to) = 0. (3.4)

The classical Sturm compar ison theorem asserts that

( T c ( t - - t ~ Z]. (3.5) u(t) <- v(t) = cos ,

Letting t = z, we obtain the inequality

to)) 0 < u, - c~ cos! . (3.6)

254 C. Bandle and M. K. Kwong ZAMP

It then follows that

z - to -< l. (3.7)

In other words, z is closer to to than to t~, and we have tl - z > l. By multiplying (3.2) with u'(t) < 0 and integrating the resulting inequal-

ity from to to t~, we obtain the estimate

-> (3.8)

Since the last term is a constant, we can choose ~ so large that

u'(T) < - ~ ~. (3.9)

On the other hand, (F-3) implies that F(t, u)/u is bounded from below, say by - k 2. In the interval to the right of t = % u(t) then satisfies the differential inequality

u"(t) - k2u(t) < O. (3.10)

We can thus compare it with the solution w(t) of

w " ( t ) - k 2 w ( t ) = O , w(z )=ul , w ' ( - c ) = - ~ . (3.11)

Again the Sturm comparison theorem gives

u(t) <- w(t), t > z, (3.12)

at least before either u(t) or w(t) becomes zero. It is easy to see that if ~ is large enough, w(t) must have a zero before reaching tl; hence for such values of 7, u(t) must have a zero before reaching tl. Applying Lemma 2 completes the proof for the case of (BVP2). (BVP3) can be regarded as a reflection of (BVP2) and so can be deduced as a corollary of what we have just proved.

To treat (BVP1), we choose an interior point t2~(t0, tl). We shoot a unique u(t) from t = te both towards the left and towards the right. Same arguments as before show that if the initial height u(t2) is chosen large enough, the solution will intersect the t-axis once in (to, t2) and once in 02, tO. Applying Corollary 2 now completes the proof. �9

4. A result on uniqueness

In the section we assume that

(F-5) any solution of (BVP2) on an interval [fi, t2] can be extended to a solution of (BVP1) on a larger interval [to, t2] with to < tl.

In [ 1] it is shown that for the simpler class of equations (1.3) and under the

Vol. 40, 1989 Semilinear elliptic problems in annuli 255

condition (1.6), uniqueness for (BVP1) over all intervals implies uniqueness for (BVP2) over all intervals. We shall extend this result to a wider class of equations under the weaker condition of superlinearity (S), which is as- sumed through this section.

Following the Kolodner-Coffman method, the uniqueness of boundary value problems over all intervals can be obtained from information concern- ing the variation of the shooting solution with respect to the shooting parameter at the endpoint. For a discussion of this method and proofs of the following two lemmas, see [6]. Although Lemma 9 does not require condition (S), Lemma 10 does. See also [3] and [7] for other uniqueness results.

Lemma 9. Let u~(t) be the solution shot out from t = to with initial height e and u~(t) intersects the t-axis at T(~) > to. Define v(t) = 8u~(t)/8~. If v(T(e)) < 0 for all e, then (BVP2) has a unique solution over every interval [to, t,].

Similarly let us(t ) be the solution shot out from t = to with initial slope /~ and urn(t) intersects the t-axis at T(/?)> to. Define w(t)= c~u~(t)/c?~. If w(T(~)) < 0 for all/~, then (BVP1) has a unique solution over every interval [to, t,].

Lemma 10. The functions v(t) and w(t) satisfy the differential equations

v"(t) + Fu(t, u )v(t) = O, (4.1)

and

w"(t) + F,(t, ul~)w(t ) = O, (4.2)

respectively. They are non-negative near to and change sign at least once in [to, T(~)], respectively [to, T(/~)].

Hence, if we know that v(t) or w(t) does not change sign more than once, then the condition in Lemma 9 is satisfied. In the case of (BPV1) we can also think of the shooting being done from the right endpoint, and use the corresponding function #(t). Since w(t) and #(t) satisfy the same differential equation (4.2), by the classical Sturm separation theorem, they change sign the same number of times. In other words, w(fi) and #(t0) have the same sign.

The condition in Lemma 9, that v(t) or w(t) is negative at the endpoint of the interval implies the fact that T(c0 and T(/?) are strictly decreasing functions of e and /~ respectively, and uniqueness follows obviously. Con- versely, if the monotonicity of T(c0 or T(/~) is violated at some point, then uniqueness fails.

256 C. Bandle and M. K. Kwong ZAMP

Lemma 11. If there exist two intervals [to, fi] and [to, fi] such that w(tO < 0 but w(fi)> 0, then uniqueness for (BVP1) cannot hold for all intervals.

Proof. If to = t0, we consider the two solutions as being shot from to. At fi, T(fl) is decreasing while at fi it is increasing and so uniqueness cannot hold. If t~ = fi, the same argument works if we consider the two solutions as being shot from tl and use ~(t). In the general case, we introduce the solution for (BVP1) on [to, fi] and repeat the argument for this solution and one of the other two. �9

Lemma 12. Suppose that there exists a point z such that

(F-6) F,(z, u)/F(z, u) is uniformly bounded for all u.

Then the function w(t) associated with the solution of (BVP1) over a sufficiently small interval [z, tr] can only change sign once.

Proof. By continuity we can choose a so close to z that for all t ~['c, a]

2F(t, u) + (t - z)F,(t, u) > O, (4.3)

and

2F(t, u) - (a -- OFt(t, u) > O. (4.4)

Inequalities of these forms have been shown to give uniqueness of (BVP1) and (BVP2) by Coffman [2] and Ni [8]. The most general result for the case in which F(t, u) is a product q(t)f(u) has been summarized in Theorems 9 and 7 in [6]. The proof extends without modifications to the more general equation (2.1), and indeed consists of showing that w(t) or v(t) cannot change sign more than once. �9

Lemma 13. Suppose that for some z, (F-6) holds, and (BVP1) has a unique solution for all intervals [to, tl]. Then the function w(t) associated with the solutions cannot change sign more than once in [to, tl).

Proof. Suppose the contrary; in some interval, w(t) changes sign more than once. The number of sign changes must be odd in order to keep T(fl) a monotonically decreasing function; so it must be at least 3. By continuity, as the interval is shrunk towards [T, a], in which (according to Lemma 12) w(t) only changes sign once, some intermediate interval must exist in which the corresponding w(t) changes sign twice. By Lemma 11, this will contradict the uniqueness of (BVP1) over all intervals. �9

Theorem 4. Suppose that conditions (S), (F-5), and (F-6) hold. If boundary value problem (2.1), (2.2a) has a unique solution over all intervals, then boundary value problem (2.1), (2.2b) (respectively (2.2c)) has a unique solution.

Vol. 40, 1989 Semilinear elliptic problems in annuli 257

Proof. Let u(t) be a given solution of (BVP2) over [to, tl]. By (F-5) we can continue it to a solution of (BVP1) over [t2, tl]. Consider the functions v(t) and w(t) associated with the two boundary value problems. In the interval [to, tl], the two differential equations (4.1) and (4.2) coincide. By Lemma 13, w(t) has only one zero in (t2, tO. Thus w(t) has either one or no zero in the smaller interval (to, h). Using (S) with the Sturm comparison theorem, we know that w(t) oscillates faster than u(t) in [to, h]. It follows that w'(to) < u'(to) = v'(to) = 0. This inequality between the starting values of w' and v' and the fact that w and v satisfy the same second order differential equation in [to, h] imply, by the Sturm comparison theorem again, that w oscillates faster than v. Since w does not have more than two zeros in [to, h], neither can v. The uniqueness of (BVP2) then follows from Lemma 9. II

References

[I] Bandle, C., Coffman, C. V., and Marcus, M., Nonlinear elliptic problems in annular domains, J. Diff. Eq., 69. 322-345 (1987).

[2] Coffman, C. V., On the positive solutions of boundary value problems for a class of nonlinear differential equations, J. Diff. Eq., 3, 92-111 (1967).

[3] Coffman, C. V., Uniqueness of the ground state solution for A u - u .4-u3= 0 and a variational characterization of other solutions, Arch. Rat. Mech. Anal., 46, 81-95 (1972).

[4] Coffman, C. V., and Marcus M., Existence and uniqueness results for semilinear Diriehlet problems in annuli, (preprint).

[5] Garaizar, X., Existence of positive radial solutions for semilinear elliptic equation in the annulus, J. Diff. Eq., 70, 6%92 (1987).

[6] Kwong, Man Kam, On the Kolodner-Coffrnan method for the uniqueness problem of Emden-Fowler BVP (preprint).

[7] Nehari, Z., On a class of nonlinear second order differential equations, Trans. Amer. Math. Soc., 95, 101-123 (1960).

[8] Ni, W. M., Uniqueness of solutions of nonlinear Dirichlet problems, J. Diff. Eq., 50, 289-304 (1983).

Abstract

The method of shooting is used to establish existence of positive radially symmetric solutions to nonlinear elliptic equations of the form Au + f ( r , u) = 0 on annular regions a < r = Ix] < b in R N, satisfying Dirichlet or Neumann conditions on the boundary. This extends recent work done by Bandle, Coffman and Marcus. A result concerning uniqueness of such solutions is also extended.

Zusammenfassung

Mit Hilfe eines Schiessverfahrens wird die Existenz von L6sungen nichtlinearer ProNeme der Form zXu +f(r, u) = 0 in ringf6rmigen Gebieten nachgewiesen, die verschiedenen Randbedingungen genfigen. Es wird auch ihre Eindeutigkeit untersucht. Diese Arbeit verallgemeinert gewisse Ergebnisse von Bandle, Coffman und Marcus.

R~sum~

On utilise une m&hode de tir pour ~tablir l'existence de solutions de probl6mes non linbaires du type Au +f(r, u) = 0 dans des anneaux, v6rifiant diff~rentes conditions aux limites. Ensuite on discute l'unicit6 de ces solutions. Ce travail g6n6ralise certains r6sultats de Bandle, Coffman et Marcus.

(Received: June 20, 1988)