Multiple Solutions for a Class of Dirichlet Double Eigenvalue Quasilinear Elliptic Systems Involving the ()-Laplacian Operator
Existence results of three weak solutions for a Dirichlet double eigenvalue quasilinear elliptic system involving the ()-Laplacian operator, under suitable assumptions, are established. Our main tool is based on a recent three-critical-point theorem obtained by Ricceri. We also give some examples to illustrate the obtained results.
1. Introduction and Preliminaries
The aim of this paper is to investigate the existence of at least three weak solutions for the following Dirichlet double eigenvalue quasilinear elliptic system: for , where is the -Laplacian operator, is a nonempty bounded open set with a boundary of class , and are positive parameters, and for . Here, are measurable functions with respect to for every and are with respect to for a.e. , and and denote the partial derivatives of and with respect to , respectively.
Moreover, and satisfy the following additional assumptions: for every and every , for a.e. ; for every and every ,
Here and in the following, we let be the Cartesian product of the Sobolev spaces for ; that is, equipped with the norm where for , Put Since for , is compactly embedded in , so that . In addition, it is known (see [1, formula (6b)]) that for , where denotes the Gamma function defined by and is the Lebesgue measure of the set , and equality occurs when is a ball.
Moreover, let Simple calculations show that there is such that , where stands for the open ball in of radius centered at .
Put for .
By a (weak) solution of system (1), we mean any such that for all .
In the literature many papers (see, e.g., the papers [2–9] and references therein) deal with nonlinear elliptic problems. Motivated by the fact that such kind of problems is used to describe a large class of physical phenomena, many authors have studied the existence and multiplicity of solutions for (1).
The goal of this work is to establish some new criteria for system (1) to have at least three weak solutions in , by means of a very recent abstract critical point result of Ricceri . We first recall the following three-critical-point theorem that follows from a combination of [11, Theorem 3.6] and [10, Theorem 1].
Lemma 1. Let be a reflexive real Banach space; let be a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on , bounded on bounded subsets of ; a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
Assume that there exists and , with , such that (a1); (a2)for each, the functional is coercive.
Then, for each compact interval , there exists with the following property: for every and every functional with compact derivative, there exists such that, for each , the equation has at least three solutions in whose norms are less than .
2. Main Result
In the present section we discuss the existence of multiple solutions for system (1). For any , we denote by the set This set will be used in some of our hypotheses with appropriate choices of .
We formulate our main result as follows.
Theorem 2. Assume that there exist two positive constants and with such that (b1) for a.e. and all for ; (b2)(b3)uniformly with respect to .
Then, setting for each compact interval , there exists with the following property: for every , there exists such that, for each , system (1) admits at least three weak solutions in whose norms are less than .
Proof. Our aim is to apply Lemma 1 to our problem. To this end, for each , we let the functionals be defined by
Clearly, is bounded on each bounded subset of and it is known that and are well-defined and continuously Gâteaux differentiable functionals whose derivatives at the point are the functionals and given by
for every , and is sequentially weakly lower semicontinuous (see Proposition 25.20 of ). Also, is a uniformly monotone operator in (for more details, see (2.2) of ), and since is coercive and hemicontinuous in , by applying Minty-Browder theorem (Theorem 26.A of ), admits a continuous inverse on .
We claim that is a compact operator. To this end, 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 ). Since is in for every , the derivatives of are continuous in for every , so fo , strongly as . By the Lebesgue control convergence theorem, strongly as . Thus we proved that is strongly continuous on , which implies that is a compact operator by [14, Proposition 26.2]. Hence the claim is true.
Moreover, we have Next, put such that for , and . Clearly and, in particular, one has for So Bearing in mind that and that for , one has .
Since for each for , condition (b1) ensures that Hence Moreover, owing to assumption (b2), we have Taking into account that for each for (see (6)), we have that for every , and taking into account (26) and (28), it follows that Therefore, assumption (a1) of Lemma 1 is satisfied.
Now, fixed , due to (b3), there exist two constants with such that for all and for all . Fix . Then for all . So, for any fixed , from (28) and (32), we have and thus which means that the functional is coercive. Then, also condition (a2) of Lemma 1 holds.
In addition, since is a measurable function with respect to for every and is with respect to for a.e. , satisfying condition (G), the functional is well defined and continuously Gâteaux differentiable on with a compact derivative, and for all , . Thus, all the hypotheses of Lemma 1 are satisfied. Also note that the solutions of the equation are exactly the weak solutions of (1). So, the conclusion follows from Lemma 1.
We now point out the following special case of Theorem 2 when does not depend on .
Theorem 3. Let be a -function and assume that there exist two positive constants and with such that (b4) for all for ; (b5)(b6) Then, settingfor each compact interval , there exists with the following property: for every , there exists such that, for each , the system for , admits at least three weak solutions in whose norms are less than .
Proof. Set for all and for . Since , Theorem 2 ensures the conclusion.
Let and . Then we have the following existence result.
Corollary 4. Let be a continuous function and let be an -Carathéodory function. Put for each and assume that there exist two positive constants and with such that (b7) for all ;(b8);(b9) Then, setting for each compact interval , there exists with the following property: for every , there exists such that, for each , the problem admits at least three weak solutions in whose norms are less than .
Now, we want to point out a simple consequence of Corollary 4 in the case when and . For simplicity, we fix and note that in this situation, and .
Corollary 5. Let be a continuous function and an -Carathéodory function. Put for each and assume that there exist two positive constants and with such that assumption (b7) in Corollary 4 holds, and (b10);(b11) Then, setting for each compact interval , there exists with the following property: for every , there exists such that, for each , the problem admits at least three classical solutions in whose norms are less than .
First, we present an example of the application of Theorem 3.
Example 1. Let . Consider the system where is an arbitrary function which is measurable with respect to for every and is with respect to for a.e. , satisfying for every and . Taking into account , picking and for each , so that and , by choosing , we have for all and We see that which gives that Hence, all the assumptions of Theorem 3, with , are satisfied. So, setting for each compact interval , there exists with the following property: for every , there exists such that, for each , system (45) has at least three weak solutions in whose norms are less than .
The next example follows directly by Corollary 5.
Example 2. Suppose for all . Then one has for all . Now, consider the following two-point boundary value problem
where is an arbitrary -Carathéodory function. Note that, in this case, , , , and . In fact, by choosing and , we have for all . Also
which shows that assumption (b10) is fulfilled. Furthermore, we have . Thus, all hypotheses of Corollary 5 are satisfied. So, setting , for each compact interval , there exists with the following property: for every , there exists such that, for each , problem (52) has at least three classical solutions in whose norms are less than .
In particular, there exist two positive constants and such that, for each , the problem admits at least three classical solutions in whose norms are less than .
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