Abstract
Using critical point theory due to Bonanno (2012), we prove the existence of at least one nontrivial solution for Dirichlet boundary value systems with the -Laplacian.
1. Introduction
In this note, we prove the existence of at least one nontrivial solution for Dirichlet boundary value systems with the -Laplacian as follows: for , where is an integer, where denotes the -Laplacian differential operator and for , , is a nonempty bounded open set with smooth boundary , is a function such that is measurable in for all , is in for every and for every , and denotes the partial derivative of with respect to for .
Due to importance of second-order Dirichlet and Neumann problems in describing a large class of physic phenomena, many authors have studied the existence and multiplicity of solutions for such a problem; we refer the reader to [1–13] and references therein. Some authors also study the system case; see [14–20].
In [6], the authors, employing a three-critical-point theorem due to Bonanno [3], determined an exact open interval of the parameter for which system (1) in the case admits non nontrivial weak solution.
The aim of this paper is to prove the existence of at least one nontrivial weak solution for (1) for appropriate values of the parameter belonging to a precise real interval, which extend the results in [7]. For basic notation and definitions and also for a thorough account on the subject, we refer the reader to [21].
2. Preliminaries and Basic Notation
First, we recall for the reader’s convenience [22, Theorem 2.5] as given in [23, Theorem 2.1] (see also [3, Proposition 2.1]) which is our main tool to transfer the question of existence of at least one weak solution of (1) to the existence of a critical point of the Euler functional as follows.
For a given nonempty set and two functionals , we define the following two functions as follows: for all , .
Theorem 1 ([3, Theorem 5.1]). Let be a reflexive real Banach space and let be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on and let be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Put and assume that there are , , such that Then, for each , there is such that and .
Let us introduce notation that will be used later. Let be the Sobolev space with the usual norm given by , where . Let with the norm where . Let Since for , one has . In addition, it is known [24, formula (6b)] that for , where denotes the Gamma function and is the Lebesgue measure of the set and equality occurs when is a ball.
Now, fix and pick with such that , where stands for the open ball in of radius centered at . Set
We say that is a weak solution to (1) if and for every . For , we denote the set
Now, we prove the following lemma to get the main result.
Lemma 2. Let be the operator defined by for every , . Then, admits a continuous inverse on .
Proof. Since
is coercive. For any , by (2.2) in [25], there exists such that
for any . Now, for any and , we have
Then, from (13), we see that, for with ,
where and . Thus, is strictly monotone. Clearly, is also demicontinuous; so, by [26, Theorem 26.A(d)], the inverse operator of exists.
Now, we show that satisfies property as follows:if and , then . Let us take a sequence such that in and
Then,
So,
According to the uniform convexity of , in .
It suffices then to show the continuity of . Let be a sequence of such that in . Let and in be such that
By the coercivity of , one deduces that the sequence is bounded in the reflexive space . For a subsequence, we have in , which implies
It follows from the property of and the continuity of that in and in .
Moreover, since is an injection, we conclude that .
3. Results
For a given nonnegative constant and a positive constant with , put where (see (10)).
We formulate our main result as follows.
Theorem 3. Assume that there exist a nonnegative constant and two positive constants and with and such that (A1) for each ;(A2). Then, for each , system (1) admits at least one nontrivial weak solution such that
Proof. To apply Theorem 1 to our problem, arguing as in [7], we introduce the functionals for each as follows: It is well known that and are well defined and continuously differentiable functionals whose derivatives at the point are the functionals , given by for every , respectively. Moreover, is sequentially weakly lower semicontinuous, admits a continuous inverse on , and is sequentially weakly upper semicontinuous (see Lemma 2). Furthermore, is a compact operator. Indeed, it is enough to show that is strongly continuous on . For this, for fixed , let weakly in as ; then, we have that converges uniformly to on as (see [26]). Since is in for every , the derivatives of are continuous in for every ; so, for , strongly as which follows strongly as . Thus, we proved that is strongly continuous on , which implies that is a compact operator by Proposition 26.2 of [26]. Set such that, for , , and . It is easy to see that and, in particular, one has, for , so, from the definition of , we have From the conditions and , we obtain Moreover, from (6), one has for each ; so, from the definition of , we observe that from which it follows Since, for , for each , condition (A1) ensures that So, one has On the other hand, by similar reasoning as before, one has Hence, from Assumption (A2), one has . Therefore, from Theorem 1, taking into account that the weak solutions of the system (1) are exactly the solutions of the equation , we have the conclusion.
Now, we point out the following consequence of Theorem 3.
Theorem 4. Suppose that there exist two positive constants and with such that (A3); (A4) for every . Then, for each system (1) admits at least one nontrivial weak solution such that .
Proof. The conclusion follows from Theorem 3, by taking and . Indeed, owing to our assumptions, one has On the other hand, taking Assumption (A4) into account, one has Moreover, since for each , an easy computation ensures that whenever . Now, owing to Assumption (A3), it is sufficient to invoke Theorem 3 for concluding the proof.
Let be a continuous function in and let , for , be a function in and
Theorem 5. Assume that Assumption (A1) in Theorem 3 holds. Suppose that there exist two positive constants and with such that (B1) for each and for ;(B2); (B3) for every for . Then, for each system admits at least one nontrivial weak solution such that .
The following result gives the existence of at least one nontrivial weak solution in to the problem (1) in the autonomous case. For a given nonnegative constant and a positive constant with , put where (see (10)). We have the following result as a direct consequence of Theorem 5.
Theorem 6. Let be a function and assume that there exist a nonnegative constant and two positive constants and with and such that (C1) for each ;(C2). Then, for each , the problem admits at least one nontrivial weak solution such that .
Proof. Note that , which together with (B2) implies Therefore, we have the conclusion by Theorem 5.
As an example, we point out the following special case of our main result.
Theorem 7. Let be a positive and essentially bounded function and let be a nonnegative continuous function such that . Then, for each , the problem admits at least one nontrivial weak solution in .
Proof. For fixed as in the conclusion, there exists positive constant such that Moreover, implies . Therefore, a positive constant satisfying can be chosen such that Hence, arguing as in the proof of Theorem 4, the conclusion follows from Theorem 6 with , , and for every .
We remark that if for every , then Theorem 7 gives [7, Theorem 1.1].
Remark 8. For fixed , put . The result of Theorem 7 for every holds with for all , where is the ensured nontrivial weak solution in (see [7, Corollary 3.1]).
We close this paper by presenting the following example to illustrate our results.
Example 9. Consider the problem where , , and . Let for all , where . It is clear that . Note that and ; we have , and so . Hence, taking Remark 8 into account, since , by applying Theorem 7, for every , the problem (48) has at least one nontrivial classical solution such that .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by Scientific Research Fund of Sichuan Provincial Education Department (no. 12ZB081).