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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 510943, 8 pages
http://dx.doi.org/10.1155/2013/510943
Research Article

Multiplicity and Bifurcation of Solutions for a Class of Asymptotically Linear Elliptic Problems on the Unit Ball

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 13 December 2012; Revised 25 January 2013; Accepted 6 February 2013

Academic Editor: Ti J. Xiao

Copyright © 2013 Benlong Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper mainly dealt with the exact number and global bifurcation of positive solutions for a class of semilinear elliptic equations with asymptotically linear function on a unit ball. As byproducts, some existence and multiplicity results are also obtained on a general bounded domain.

1. Introduction

In this paper, we are concerned with positive solutions of the following elliptic equation subject to homogeneous Dirichlet boundary condition where is a smooth bounded domain in , is a positive parameter, , and the function satisfies the following.(F1) is a positive function, and is strictly convex; that is, is strictly increasing in .(F2) is asymptotically linear, that is,

For the past years, this problem attracted attentions of many authors. It was studied in [14] with being strictly increasing and was studied in [57] with a specific function which is not increasing.

The main goal of this paper is to study the exact number and bifurcation structure of the solutions of on a unit ball , with a general asymptotically linear function . Some results in this paper (see Section 3) can be viewed as an extension and improvement of that in [7], but the argument approach here is very different to that in [7]. As byproducts, we also get some new results which also hold for general domain (see Section 2). The paper is organized as follows. In Section 2, we study the existence and multiplicity of solutions for problem on a general bounded domain, with some new results complementing those existing in the literature. In Section 3, we study the exact number and global bifurcation structure of positive solutions of on a unit ball.

2. Multiplicity of Positive Solutions on a General Domain

Throughout this section, we assume that is a smooth bounded domain in , and satisfies (F1) and (F2). We also note that, by maximum principle, all solutions of are positive on .

Before the statement of our main result, we derive some preliminary lemmas. Though some of them may be known, we provide their proofs for reader’s convenience.

Lemma 1. For any , is solvable.

Proof. Consider the functional where .
From (F1) and (F2), it is easy to see that so
Poincàre’s inequality , and the imbedding theorem of to yield so as , where , and then is coercive and bounded from below. It is also easy to see that is weakly lower semi-continuous [8, page 446, Theorem 1]. By applying direct variational methods [9, page 4, Theorem 1.2], we can get the desired result; that is, is reached at some point , and is a solution of when .

Lemma 2. For any , has no solution, where .

Proof. If not, assume that is a solution of for some . Multiplying by , the normalized positive eigenfunction with respect to the first eigenvalue of subject to homogenous Dirichlet boundary condition, and then integrating by parts, we get which is a contradiction.

We begin by show the following.

Lemma 3. There exists a number , such that has at least a solution for and has no solution for .

Proof. Let By Lemmas 1 and 2, . We need just to prove that if has a solution, then also has a solution for all . This can be done by a simple argument of sub-sup solution method, since it is easy to see that any solution of is a super solution of and a subsolution.
It is easy to see that is a subsolution of , then a standard sub-super solution method’s argument and comparison theorems give the following lemma.

Lemma 4. If is solvable, then one has a minimal solution , that is, for any solution of , . Moreover, is increasing with respect to .

Lemma 5. If , then the solution of   is unique.

Proof. Suppose that and are solutions of . Let , then By mean value theorem, satisfies where lies between and . Multiplying and integrating, we get which implies that . The proof is complete.

Lemma 6. The minimal solution is stable, that is, , where denotes the first eigenvalue of the following problem:

Proof. Suppose on the contrary that , and is the corresponding eigenvector. Let , then by and (11), we have when is small enough, and hence is a super solution of problem . On the other hand, is a subsolution of , and Hopf’s boundary lemma implies that for small. An application of sub-sup solution method guarantees that there is a solution of satisfying in , which is a contradiction with the minimality of . The proof is complete.

Now we state our main result.

Theorem 7. Suppose that satisfies (F1) and (F2), then there exists (where ) such that problem (i) has at least one solution for and a unique solution for ;(ii) has no solution for ;(iii)(a) if , then problem has no solution at , and for all , where denotes the unique solution of for (see Figure 1), (b) if , then problem has a unique solution for and , has at least two solutions for (see Figure 2 for a minimal diagram).

510943.fig.001
Figure 1: Diagram for .
510943.fig.002
Figure 2: Minimal diagram for .

Proof. Statement (i) follows from Lemmas 3 and 5. Statement (ii) follows from Lemma 3. Now we give the proof of statement (iii).
(a) Suppose . The solution bifurcates at infinity near (see [2, 10] for details). On the other hand, has a unique solution for , and no solution for . Therefore the bifurcation curve from infinity is on the left of , and hence for all by the expression of the bifurcation solution in Theorem 13 in Section 3.
If has a solution, let denote the minimal solution of . By Lemma 4, for , contradicting .
(b) For clarity, the proof will be divided into 3 steps.
Step 1. The existence and uniqueness of solutions of for .
The existence follows directly from Lemma 4. Note that , and the uniqueness can be proved in a similar way as in the proof of Lemma 5.
Step 2. The existence and uniqueness of solutions of for .
By Lemmas 3 and 4, has a minimal solution for any , and is increasing in . Let be any sequence such that . Firstly we insure that case ( ) is bounded. Suppose the contrary that . Let and , then
Since is bounded in , it follows from (13) that is bounded in . Then subject to a subsequence, we may suppose that there exits , such that Then by letting , we get from (13) in the weak sense that with , and by strong maximum principle. Hence , that is, , a desired contradiction.
Now in a similar way, the boundedness of in implies that is bounded in . Then subject to a subsequence, we may suppose that there exits , such that Then by letting , we get and the existence is proved.
Now we prove the uniqueness. Let be the minimal solution of and a different solution. Then satisfies where satisfying . It follows that , where denotes the first eigenvalue of the operator subject to the Dirichlet boundary condition, as defined in Lemma 1. Since in , we have that , which implies that the operator is nondegenerate. Then by the Implicit Function Theorem, the solution of forms a cure in a neighborhood of , which is clearly contradicted to the definition of in (7).
Step 3. Prove that has at least two solutions for .
Following the argument in [5], we prove it by variational method of Nehari type (see [11]). As we have known (Lemma 5), there exists a minimal solution of when . Now we must look for another solution . Assuming that , with , then satisfies For convenience, let and , then we have Define and the solution manifold
Firstly we show that for any . Let be the first eigenfunction of in subject to Dirichlet boundary condition and , then
It follows from (23) that for sufficiently large if .
On the other hand, let be the eigenfunction with of the first eigenvalue of the following equation:
Since is the minimal solution, it follows from Lemmas 4 and 6 that . Then Hence when is small enough. Now it is easy to see that is not empty. In fact, take for some large , and for some small , such that respectively. Define a continuous function on , namely, Then , , and hence there exist such that , that is, , and , a desired conclusion.
Since is convex, is convex with respect to such that
Integrating (29) with respect to from 0 to , we get Therefore, on that is, is bounded from below.
And then we obtain a nonminimal positive solution of by using the Nehari variational method. The proof is complete.

Remark 8. The solutions that we get from the above discussion are weak ones, but a standard elliptic regularity argument shows that they are indeed classical solutions.

In view of Theorem 7, we want to know what conditions ensure that or . Following [4], we consider the function . It is easy to see that is strictly increasing, and hence exists (may be ). Also note that .

Theorem 9. If , then ; if , then .

Proof. (i) If , then for all . We prove that has no solution and hence . Suppose the contrary that is a solution for , then Let be a positive eigenfunction of the first eigenvalue of on with Dirichlet boundary condition, that is Multiplying (32) by , and integrating by parts, we get which yields that , contradicting the fact that .
(ii) If , we prove that .
Let be the bifurcation curve as described in Theorem 13 in Section 3, then
It follows from (33) and (35) that By the fact that ( ) a.e. in , we have for sufficiently large. It follows from (36) that when is sufficiently large, which means that the bifurcation curve from infinity is on the right of , and hence by the definition of in (7). The proof is complete.

Now we define another function which is also crucial in studying exact multiplicity in the next section. Let then a.e. in , and is strictly increasing, and . Denote

Theorem 10. If , then ; if , then .

Proof. If , then for all . It follows that is strictly decreasing and hence , which implies that .
On the other hand, if , by we get that . Then the conclusion follows for Theorem 9.

3. Exact Number and Global Bifurcation of Solutions on a Unit Ball

From Theorem 7, the exact number of solutions is now clear in the case of ; that is, the solution is unique if it exists. On the other hand, it is far from known in general exactly how may solutions of for if . Using the bifurcation approach developed in [1214], and also the idea and techniques developed in [7], we solve this problem on the unit ball under some conditions.

Throughout this section, we suppose that is the unit ball in centered with the origin.

The next remarkable results regarding are due to Gidas et al. [15] and Lin and Ni [16].

Lemma 11. (1) If is locally Lipschitz continuous in , then all positive solutions of are radially symmetric, that is, , , and satisfies Moreover, for all , and hence .
(2) Suppose . If is a positive solution to , and is a solution of the linearized problem (43) (if it exists), then is also radially symmetric and satisfies

The next lemma also plays a key role in this section.

Lemma 12. (1) For any , there is at most one such that have a positive solution with and .
(2) Let have a positive solution with   , then is open; is a well-defined continuous function from to .

Lemma 12 is well known; see, for example, [13, 17, 18]. A simple proof of the first part of the lemma can be found in [14]. Because of Lemma 12, we call the phase space, the bifurcation curve, and the phase space with bifurcation curve the bifurcation diagram.

We will also need the following theorem of bifurcation from infinity.

Theorem 13 (see [10, 19]). Suppose . Let and . Then all positive solutions of near have the form of for and some , where is a positive eigenfunction of the first eigenvalue of on subjected to Dirichlet boundary condition, , and as .

To make bifurcation argument work, a crucial thing is the following result.

Let be a solution of problem , then is called a degenerate solution if the corresponding linearized equation has a nontrivial solution.

Now suppose that satisfies (F1), (F2). As in the end of Section 2, let If , then there exists a unique real number , such that

Lemma 14. Suppose that . If is a degenerate solution of , then .

Proof . Suppose the contrary that , then Let be a nontrivial solution of the corresponding linearized equation (43). From and (43), we get It appears from (46) and (47) that must change sign in .
In view of Lemma 11 , we suppose that is a maximal zero in . We may also suppose that , for all . Then where denotes the ball of radius centered with the origin.
On the other hand, using integration by parts, we have a contradiction.

Theorem 15. Suppose that satisfies (F1)-(F2) with . If is a degenerate solution of , then any nontrivial solution of the corresponding linearized equation (43) does not change sign in .

Proof . By Lemma 14, . In view of Lemma 11, there exists , such that . Let be a non-trivial solution of the corresponding linearized equation (43), then .
We assert that has no zeroes on . Suppose the contrary and let be the largest zero of on . We may suppose that in . Note that for , a similar argument as in the proof of Lemma 14 yields a contradiction.
Now we prove that has no zeroes on . Suppose the contrary and let be the smallest zero of on . We may suppose that in . Multiplying by , (43) by , subtracting, and integrating on , we get
Let , then , , and for . Hence for . Then
On the other hand, by Green formula, A contradiction occurs from (50), (51), and (52). Hence has no zeroes in , that is to say, does not change sign in . The proof is complete.

Now define , by then the linearized operator (Frechèt derivative) is

From the maximum principle, all solutions of are positive on . Moreover, if is degenerate solution of , then by Theorem 15, the nontrivial solution of (43) does not change sign in , and hence can be chosen to be positive. Then by Krein-Rutman’s Theorem, , and it follows from Fredholm alternative theorem that codim . Now we prove that . If it is not the case, then there exists , such that

We also have

Multiplying (55) by , (56) by , subtracting, and integrating, we obtain a contradiction. As all the conditions of Crandall-Rabinowitz’s bifurcation theorem [20] are satisfied, the solutions of near the degenerate solution form a smooth curve which is expressed in the form where is a smooth function near with , where is a complement of span in , and is the positive solution of (43), which is unique if normalized.

Substituting and by expression (58), then differentiating the equation twice, and evaluating at , we have Multiplying (59) by , (43) by , subtracting, and integrating, we obtain

By (60) and the Taylor expansion formula of at , we conclude that at any degenerate solution of , the solution curve turns left, that is to say, there is no any solution on the right near . This observation is very important to our proof of the following theorem.

Theorem 16. Suppose that is the unit ball in , satisfies (F1)-(F2), and . Then for problem , (1)there exist no solutions for , (2)there exists exactly one solution for , (3)there exist exactly two solutions for .
Moreover, the solution set of forms a smooth curve in the space , which can be roughly described as in Figure 3.

510943.fig.003
Figure 3: Precise bifurcation diagram on a unit ball.

Proof. By Theorem 10, , and Theorem 7 tells us that has a unique solution for , and Implicit Function Theorem implies that is a degenerate solution. By Theorem 15, non-trivial solution of the corresponding linearized equation (43) does not change sign in , and we may suppose that is positive in . Then Crandall-Rabinowitz’s bifurcation theorem [20] and the discussion prior to this theorem imply that the solutions near form a smooth curve which turns to the left in the phase space. We may call the part of the smooth solution curve with the upper branch, and the rest the lower branch. We denote the upper branch by and the lower branch by .
For the upper branch, as long as nondegenerate, the Implicit Function Theorem ensures that we can continue to extend this solution curve in the direction of decreasing . We still denote the extension by . This process of continuation towards smaller values of will not encounter any other degenerate solutions. This is because, if, say, becomes degenerate at , the discussion prior to this theorem implies that all the solutions near must lie to the left side of it, which is a contradiction. Lemma 12 tells us that is decreasing. So in the progress of extension of towards smaller values of , there are only the following two possibilities.(i)The upper branch stops at some , and . (ii) goes to infinity as ,   .
But case (i) cannot happen, since is obviously not a solution of   . Hence case (ii) happens. We assert that . In fact, let be an arbitrary sequence such that . Denote , , then and Since is bounded, by Sobolev Imbedding Theorems and standard regularity of elliptic equation, it is easy to see that has a subsequence, still denoted by , such that in , for some , in . Letting in (61), we get which implies that .
Now we study the structure of the lower branch. As in the case of upper branch, as long as nondegenerate, the Implicit Function Theorem ensures that we can continue to extend this solution curve in the direction of decreasing . We still denote the extension by . This process of continuation towards smaller values of will not encounter any other degenerate solutions. Lemma 12 implies that is increasing. So in the progress of extension of towards smaller values of , there are only the following two possibilities.(i)The lower branch stops at some with .(ii)The lower branch stops at some with .
As before, case (i) will not happen. Then case (ii) happens. By , it is easy to see that . That is to say, the lower branch of solutions extends till the origin in the phase plane.
By the above argument, we obtain a smooth positive solution curve which consists of an upper branch and a lower branch . The lower branch starts from and stops at , and is a strictly increasing function. The upper branch starts from and stops at , and is a strictly decreasing function with blowing up as . By Lemma 12, all solutions of are contained in this smooth solution curve, and the complete bifurcation diagram can be described as in Figure 3. The proof is complete.

Acknowledgment

The author thanks the reviewer for his/her comments which helped to improve the content of the paper.

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