Global Bifurcation of Positive Solutions of Asymptotically Linear Elliptic Problems

Abstract

We are concerned with determining values of , for which there exist positive solutions of the nonlinear elliptic problem , The proof of our main results is based upon unilateral global bifurcation theorem of López-Gómez.

1. Introduction

Let be a bounded domain of Euclidean space , , with smooth boundary . In this paper, we consider the nonlinear elliptic boundary value problem where , is a parameter, and is the unit exterior normal to .

We make the following assumptions.(H0) with in ; with and on .(H1) is an odd function with for and there exist constants and functions , such that (H2) is an odd function with for and there exist constants and functions , such that

A solution of (1) is said to be positive if on . The purpose of this paper is to study the global bifurcation of positive solutions for the asymptotically linear elliptic eigenvalue problems (1).

Let be the space of continuous functions on . Then, it is a Banach space with the norm Let Then is a cone which is normal and has a nonempty interior and . Moreover,

By a constant we denote the first eigenvalue of the eigenvalue problem By a constant we denote the first eigenvalue of the eigenvalue problem It is well known (cf. Krasnosel’skii [1]) that, for , is positive and simple and that it is a unique eigenvalue with positive eigenfunctions . In what follows, the positive eigenfunction is normalized as .

Let be the closure of the set is a positive solution of (1) in .

Theorem 1. Let (H0)–(H2) hold. Then, there exists an unbounded, closed, and connected component in , which joins with .

Corollary 2. Let (H0)–(H2) hold. Assume that either or Then (1) has at least one positive solution.

Remark 3. Ambrosetti et al. [2] and Umezu [3, 4] only studied the bifurcation from infinity for nonlinear elliptic eigenvalue problems. Nonlinear eigenvalue problems of ordinary differential equations have been extensively studied by many authors via fixed point theorem in cones and bifurcations techniques; see Henderson and Wang [5] and Ma [6, 7] and the references therein. Ma and Thompson [7] considered the two-point boundary value problem By using the well-known Rabinowitz global bifurcation theorem [8], they proved the following.

Theorem A (see [7, Theorem ]). Assume that(A1) is continuous and on any subinterval of ;(A2) with for ;(A3)there exist such that and ;(A4) is the th-eigenvalue of , .If either or , then (12) has two solutions and such that has exactly zero in and is positive near , and has exactly zero in and is negative near .

Obviously, Corollary 2 is a higher dimensional analogue of Ma and Thompson [7, Theorem 1.1] with .

Remark 4. Shi [9] studied the exact number of all nontrivial solutions for for in certain parameter range. He proved the existence of global smooth branches of positive solutions by using the implicit function theorem under some further restrictions on .

Remark 5. Nonlinear elliptic eigenvalue problems have been studied in [4, 10] via topological degree and global bifurcation techniques. The positone case is considered in [10], which is extended to the semipositone case in [4]. An emphasis is, in Theorem 1 and Corollary 2, no assumption imposed on the boundedness of the function in (3).

Remark 6. Precup [11] applied the Moser-Harnack inequality for nonnegative superharmonic functions to produce a suitable cone and developed fixed point theorem in cones of Krasnoselskii-type to discuss the existence and multiplicity of positive solutions to elliptic boundary value problems The constant in [11, (3.1)] and the constant in [11, (3.2)] are not optimal so that [11, Theorem 3.1] is not sharp. However, (10) and (11) in Corollary 2 are optimal. In fact, for the function which satisfies , the elliptic problem has no positive solution.

The rest of this paper is organized as follows. The proof of our main results is based upon the unilateral global bifurcation theorem of López-Gómez, which is different from the topological degree arguments used in [2–4, 10]. So, in Section 2, we state a preliminary result based upon unilateral global bifurcation theorem of López-Gómez. In Section 3, we reduce (1) into a compact operator equation. Section 4 is devoted to the proof of Theorem 1.

2. Unilateral Global Bifurcation Theorem of López-Gómez

Let be a Banach space with the normal . Let stand for the space of linear continuous operators in . Let . Let be a nonlinear operator of the form where is a continuous operator compact on bounded sets such that as uniformly in any compact interval of , is a linear compact operator, and is a simple characteristic value of ; that is, for some satisfying

Let be the closure of the set Let (resp., ) be the component of that meets and around lies in (resp., ); see [12, Section 6.4] for the details.

Let Then we present the unilateral global bifurcation theorem of López-Gómez; see [12, Theorem 6.4.3].

Lemma 7 (see [12], unilateral global bifurcation of López-Gómez). Assume is discrete, satisfies (19), and the index changes sign as crosses . Then, for , the component satisfies one of the following:(i) is unbounded in ;(ii)there exists such that ;(iii) contains a point where is the complement of in .

3. Reduction to a Compact Operator Equation

To establish Theorem 1 we begin with the reduction of (1) to a suitable equation for compact operators. According to Gilbarg and Trudinger [13], let be the resolvent of the linear boundary value problem By Amann [14, Theorem 4.2], is uniquely extended to a linear mapping of compactly into and it is strongly positive, meaning that on for any with the condition that and on .

Let be the resolvent of the linear boundary value problem According to Amann [15, Section 4], is uniquely extended to a linear mapping of compactly into . By the standard regularity argument, problem (1) is equivalent to the operator equation Here is the usual trace operator.

Similarly, let be the resolvent of the linear boundary value problem Then is uniquely extended to a linear mapping of compactly into and it is strongly positive.

Let be the resolvent of the linear boundary value problem Then is uniquely extended to a linear mapping of compactly into . Furthermore, (1) is equivalent to the operator equation

4. The Proof of Main Results

Obviously, (H1) and (H2) imply that

Let Then and are nondecreasing and there exist such that Indeed, for any , there exists such that . Additionally, if we assume for some , then it follows that as . On the other hand, we assume that there exists such that ; then, it follows from (31) that as , as desired.

We consider as a bifurcation problem from the trivial solution .

Define then, is a strongly positive linear operator on . It is easy to verify that is completely continuous. From [14, Theorem 3.2], it follows that Define by then, we have from (30) that locally and uniformly in .

It is very easy to check that (34) enjoys the structural requirements for applying the unilateral global bifurcation theory of [12, Sections 6.4, 6.5] (by a counter example of Dancer [16], the global unilateral theorem of Rabinowitz [8] is false as stated. So, it cannot be used). As the theorem of Crandall and Rabinowitz [17] is applied to get the local bifurcation to positive solutions from , the algebraic multiplicity of Esquinas and López-Gómez [18] (see [19, Chapter 4]) equals and, therefore, by [12, Theorem 5.6.2] or [19, Proposition 12.3.1], the local index of as a fixed point of changes sign as crosses . Therefore, it follows from Lemma 7 that there exists the component that satisfies one of the following:(i) is unbounded in ;(ii)there exists with and which changes its sign on , such that (iii) contains a point where is the complement of in .

In what follows, we will show that the above Case (ii) and Case (iii) do not occur.

In fact, if is a nontrivial solution of (34), then satisfies the problem that is to say, satisfies the linear problem where

We claim that

Suppose, on the contrary, that with and . Then, there exists a sequence with for , such that and consequently Combining this with the fact that is a nontrivial solution of (34) and using the strong maximum principle [20, Theorem 2.4] and (42), it concludes that contracting .

Similarly, for with and , we get the desired contradiction.

Therefore, the claim (44) is true.

Since and are odd in , is a solution of (34) if and only if is a solution of (34). Combining this and (44) and using the fact that the eigenfunctions corresponding to the eigenvalue of the operator with have to change its sign in , it concludes that Case (ii) and Case (iii) cannot occur. Therefore, there exists an unbounded connected subset of the set such that .

Proof of Theorem 1. It is clear that any solution of (34) of the form yields a solution of (1). We will show that joins to .
Let satisfy We note that , for all since , is the only solution of (34) (i.e., (26)) for .
In fact, suppose on the contrary that is a nontrivial solution of the problem then, satisfies the linear problem where . Together with (H2) and the results of Krasnosel’skii [1], it follows that in , which is a contradiction. Therefore, (34) (i.e., (26)) with has only trivial solution.
Case  1 . In this case, we show that We divide the proof into two steps.
Step  1. We show that if there exists a constant number such that then joins to .
From (53), we have . We divide the equation by and set . Since is bounded in , choosing a subsequence and relabeling if necessary, we see that for some with . Moreover, from (31) and the fact that and are nondecreasing together with the assertion , there exists some such that and . It follows that as . Subsequently, we have since and . Therefore, where , again choosing a subsequence and relabeling if necessary.
Thus Since and , the strong positivity of ensures that Thus , and accordingly, joins to .
Step  2. We show that there exists a constant such that , for all .
From (H1) and (H2), there exist constants such that
By the same method to define and in Section 3, we may define and as follows.
Let be the resolvent of the linear boundary value problem Then, is uniquely extended to a linear mapping of compactly into and it is strongly positive. Let be the resolvent of the linear boundary value problem Let be the eigenvalue of the linear problem and let be the corresponding eigenfunction. Then Since satisfies (1) and satisfies (63), it follows from (59) and Green’s formula that and here is the surface element of . Subsequently, . Therefore, the component joins to .
Case  2 . In this case, if is such that then and moreover, Assume that there exists , such that, for all , Applying a similar argument to that used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that Again joins to and the result follows.

Proof of Corollary 2. It is a directly desired consequence of Theorem 1.

Conflict of Interests

The authors declare that they have no competing interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (no. 11361054 and no. 11201378), SRFDP (no. 20126203110004), and Gansu provincial National Science Foundation of China (no. 1208RJZA258).