`International Journal of Differential EquationsVolume 2011, Article ID 801706, 13 pageshttp://dx.doi.org/10.1155/2011/801706`
Research Article

On the Weak Solution of a Semilinear Boundary Value Problem without the Landesman-Lazer Condition

Department of Mathematics and Statistics, University of Uyo, Akwa Ibom, 520003 Uyo, Nigeria

Received 4 August 2011; Accepted 15 October 2011

Abstract

We prove the existence of weak solution to a semilinear boundary value problem without the Landesman-Lazer condition.

1. Introduction

We consider the nonlinear boundary value problem where is open and bounded, , is a simple eigenvalue of corresponding to the eigenvector , and the nonlinearity satisfies the following conditions:

Landesman and Lazer [1] considered the problem (1.1)-(1.2) with continuous function satisfying , where exist and are finite. The authors showed that if is an eigenfunction corresponding to , and , then the necessary and sufficient condition for the existence of weak solution of (1.1)-(1.2) is that The condition (1.3) is the well-known Landesman-Lazer condition, named after the authors. The result of the paper [1] has since been generalized by a number of authors which include [29], to mention a few.

We mention, briefly, few works without the assumption of the Landesman-Lazer condition. The perturbation of a second order linear elliptic problems by nonlinearity without Landesman-Lazer condition was investigated in [10]. The function was assumed to be a bounded continuous function satisfying The nonhomogeneous term was assumed to be an -function orthogonal to an eigenfunction in , which corresponds to a simple eigenvalue . Ha [11] considered the solvability of an operator equation without the Landesman-Lazer condition. The author used a nonlinear Carathéodory function which satisfies the conditions for almost all and all , where . The solvability of the operator equation is proved under some hypotheses on . The nonhomogeneous term was assumed to be an -function. Iannacci and Nkashama proved existence of solutions to a class of semilinear two-point eigenvalue boundary value problems at resonance without the Landesman-Lazer condition, by imposing the same conditions as in [11] in conjunction with some other hypotheses on and . Furthermore, the existence of solution was proved only for the eigenvalue . Assuming a Carathéodory function with some growth restriction and assuming an -function , Santanilla [12] proved existence of solution to a nonlinear eigenvalue boundary value problem (for eigenvalue ) without Landesman-Lazer condition. Du [13] proved the existence of solution for nonlinear second-order two-point boundary value problems, by allowing the eigenvalue of the problem to change near the eigenvalues of of the problem . The author did not use the Landesman-Lazer condition and imposed weaker conditions on than in [12]. Recently, Sanni [14] proved the existence of solution to the same problem considered by Du [13] with exactly, without assuming the Landesman-Lazer condition. The author assumed that and . Other works without the assumption of Landesman-Lazer condition include [1521]. We mention that most of the papers on this topic use the methods in [22] and [12]. The method of upper and lower solutions is used in [14]. For several other related resonance problems, we refer the reader to the book of Rădulescu [23].

The current work constitutes further deductions on the problem considered by Landesman and Lazer [1] and is motivated by previous works and by asking if it is possible to obtain a weak solution of (1.1)-(1.2) by setting . The answer is in the affirmative. The substitution gives rise to a degenerate semilinear elliptic equation. Consequently, we prove the existence of weak solution to the degenerate semilinear elliptic equation in a -weight Sobolev’s space, by using the Schaefer’s fixed point theorem. For information on weighted Sobolev’s spaces, the reader is referred to [24, 25]. The current work is significant in that the condition enables a relaxation of the Landesman-Lazer condition (1.3), and the solution to (1.1)-(1.2) is constructed using the eigenfunctions . Furthermore, the current analysis takes care of the situation where .

The remaining part of this paper is organized as follows: the weighted Sobolev’s spaces used are defined in Section 2. In addition, we use the substitution to get the degenerate semilinear elliptic equation in , from which we give a definition of a weak solution. Furthermore, we state two theorems used in the proof of the existence result. In Section 3, we prove the existence and uniqueness of solution to an auxiliary linear problem. In Section 4, we prove a necessary condition for the existence of solution to (1.1)-(1.2) before proving the existence of solution to (1.1)-(1.2). At the end of Section 4, we prove that is in , provided that . Finally, we give an illustrative example in Section 5 for which our result applies.

2. Preliminaries

We define the following weighted Sobolev’s spaces used in this paper: where . where .

For brevity, we set .

Set in (1.1) to deduce Note that the first term on the left of (2.3) vanishes, multiply (2.3) by and use (1.2) to deduce Thus, if we can prove the existence of solution to (2.4), then solves (1.1)-(1.2). Indeed, we will prove that the solution belongs to the Sobolev space .

Definition 2.1. We say that is a weak solution of the problem (2.4) provided for each .

Definition 2.2. Let be a Banach space and a nonlinear mapping. is called compact provided for each bounded sequence the sequence is precompact; that is, there exists a subsequence such that converges in (see [26]).
The following theorems are applied in this paper.

Theorem 2.3 (Bolzano-Weierstrass). Every bounded sequence of real numbers has a convergent subsequence (see [27]).

Theorem 2.4 (Schaefer’s Fixed Point Theorem). Let be a Banach space and a continuous and compact mapping. Suppose further that the set is bounded. Then has a fixed point (see [26]).

3. Auxiliary Linear Problem

Consider the linear boundary value problem: where is a strictly positive constant; , , and are functions of only.

Theorem 3.1 (a priori estimates). Let be a solution of (3.1)-(3.2). Then and we have the estimate for some appropriate constant .

Proof. Multiply (3.1) by , integrate by parts and apply (3.2) to get Using , the second term in the bracket on the right side of (3.4) may be estimated as Simplifying (3.5), we deduce (see [26]) for some constant . Notice that (3.6) implies that so that .
Using (3.7) and choosing sufficiently small in (3.4) and simplifying, we deduce (3.3).

Definition 3.2. (i) The bilinear form associated with the elliptic operator defined by (3.1) is for ,
(ii) is called a weak solution of the boundary value problem (3.1)-(3.2) provided for all , where denotes the inner product in .

Theorem 3.3. satisfies the hypotheses of the Lax-Milgram theorem precisely. In other words, there exists constants such that(i), (ii), for all .

Proof. We have for appropriate constant . This proves (i).
We now proof (ii). We readily check that for some constant . We can for example take .

Theorem 3.4. There exists unique weak solution to the degenerate linear boundary value problem (3.1)-(3.2).

Proof. The hypothesis on and (3.7) imply that . For fixed , set for all (where denotes the pairing of with its dual). This is a bounded linear functional on and thus on . Lax-Milgram theorem (see, e.g., [26]) can be applied to find a unique function satisfying for all . Consequently, is the unique weak solution of the problem (3.1)-(3.2).

4. Main Results

Theorem 4.1. The necessary condition that be a weak solution to (1.1)-(1.2) is that

Proof. Suppose is a weak solution of (1.1)-(1.2). For a test function , using integration by parts, we have: from which (4.1) follows, since .

Theorem 4.2. Let the condition (4.1) of Theorem 4.1 holds. Then there exists a weak solution to the problem (2.4).

Proof. The proof is split in seven steps.
Step 1. A fixed point argument to (2.4) is Define a mapping by setting whenever is derived from via (4.3). We claim that is a continuous and compact mapping. Our claim is proved in the next two steps.Step 2. Choose , and define . For two solutions of (4.3), we have Using (4.5), we obtain an analogous estimate to (3.4), namely: Now using the condition (). We may now use (4.7) in (4.6) and simplify to deduce for some constant . Thus, the mapping is Lipschitz continuous, and hence continuous.Step 3. Let be a bounded sequence in . By Bolzano-Weierstrass theorem, it has a convergent subsequence, say . Define Using (4.8)-(4.9), we deduce Thus, in . Therefore, is compact.Step 4. Define a set . We will show that is a bounded set. Let . Then for some . Thus, we have . By the definition of the mapping , is the solution of the problem Now, (4.11) are equivalent to Using (4.12) we have an analogous estimate to (3.3) of Theorem 3.1, namely: Choosing sufficiently small in (4.13) and simplifying, we conclude that for some constant . Equation (4.14) implies that the set is bounded, since was arbitrarily chosen.
Since the mapping is continuous and compact and the set is bounded, by Schaefer’s fixed point theorem (see, e.g., [26]), the mapping has a fixed point in .
Step 5. Write . For , inductively define to be the unique weak solution of the linear boundary value problem Clearly, our definition of as the unique weak solution of (4.15)-(4.16) is justified by Theorem 3.4. Hence, by the definition of the mapping , we have for Since has a fixed point in , there exists such that Step 6. Using (4.15)-(4.16), we obtain an analogous estimate to (3.3), namely: for some appropriate constant . Using (4.18), we take the limit on the right side of (4.19) to deduce that Equation (4.20) implies the existence of a subsequence converging weakly in to .
Furthermore, using (3.7), we deduce Again, we use (4.18) to obtain the limit on the right side of (4.21) to deduce that Equation (4.22) implies the existence of a subsequence converging weakly in to in .
Step 7. Finally, we verify that is a weak solution of (2.4). For brevity, we take the subsequences of the last step as and . Fix . Multiply (4.15) by , integrate by parts and apply (4.16) to get Using the deductions of the last step, we let in (4.23) to obtain from which canceling the terms in , we obtain (2.5) as desired.

Theorem 4.3. Let be the solution of (3.1)-(3.2). Then, the solution of (1.1)-(1.2) belongs to , and we have the estimate for some constant .

Proof. We split the proof in two steps.
Step 1. Recall that satisfies the equations: Multiplying (4.26) by , integrating by parts and applying (4.27) we compute by Cauchy’s inequality with . Choosing sufficiently small in (4.29) and simplifying, we deduce for some constant .Step 2. We have for some constant . Thus, . Hence, by a Sobolev’s embedding theorem (see [26, page 269]), we have that , since .

5. Illustrative Example

Consider the following special case for : In this case, the eigenfunction , , and . Clearly is Lipschitz continuous and . Provided the necessary condition is satisfied; Theorems 4.2 and 4.3 ensure the existence of a solution of the problem (5.1). Now, the problem (5.1) admits the solution Using (5.3) in (5.2), it is not difficult to verify that the necessary condition is satisfied.

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