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Nontrivial solutions for a quasilinear elliptic system
Boundary Value Problems volume 2014, Article number: 34 (2014)
In this paper, we deal with the existence of three nontrivial solutions for the potential system of p-Laplacian equations with homogeneous Dirichlet boundary conditions. Applying the Nehari procedure and abstract linking theorem on a product space, we give a linking structure for our variational problem, and then, combining with the classical minimax principle, we obtain three nontrivial critical values for the relevant energy functional.
In this paper, we are concerned with the multiplicity of nontrivial solutions for the following system of quasilinear equations:
where (), Ω is a bounded smooth domain in (), , are Carathéodory functions (), and there exists a function such that
In recent years, many authors have studied the existence of nontrivial solutions for Laplacian systems and p-Laplacian systems, see [1–5] and the references therein. Usually the authors change the problem into the critical point problem of the corresponding energy functional and then apply the critical point theory or the variational method, or they change it into the fixed point problem of the corresponding compactly continuous mapping and then apply topological degree theory or the method of lower and upper solutions. For instance, in  Costa and Magalhaes unified the cooperative and noncooperative Laplacian systems, and they got the existence of nontrivial solutions via the variational approach; in  Conti et al. dealt with the competitive Laplacian system, and they established the existence of positive solutions by the Nehari procedure, critical point theory, and topological degree theory; in [3, 4] the authors studied the sublinear p-Laplacian systems, and they obtained the existence of positive solutions by the method of lower and upper solutions and Leray-Schauder degree theory, respectively; in  Zhang and Zhang considered the existence of nontrivial solutions for nonlinear Laplacian systems and p-Laplacian systems applying the direct variational method. More recently, some authors discussed the multiplicity of nontrivial solutions for Laplacian systems and p-Laplacian systems, see [6–12] and references therein. In [6, 11, 12], the authors provided the existence results of three nontrivial solutions for system (1.1) with one parameter for the case , where the main methods used are the Nehari procedure and the linking theorem on product space. In  Motreanu and Zhang consider a general noncoercive quasilinear elliptic system, they establish the existence of two opposite constant sign solutions; in the case where the system has a variational structure, under the proper hypotheses, they obtain a third nontrivial solution, which is sign changing in the sense that one cannot have both components of the new solution of the same constant sign; their approach relies on a suitable method of sub-supersolutions combined with truncation and variational arguments that do not require a subcritical growth condition. In  Shen and Zhang established the existence of two positive solutions for multi-parameter p-Laplacian systems with critical exponents by use of the Nehari procedure and the variational approach. In [7, 10], the present author and coauthors dealt with a class of Laplacian systems with superlinear and sublinear terms applying the fixed point index formula on a product cone, and they obtained the existence and multiplicity of positive solutions.
Motivated by some ideas in [1, 7, 9, 12], we shall deal with the existence of nonnegative solutions (especially, positive solutions) for system (1.1) with superlinear and subcritical nonlinear terms. It is well known that the Ambrosetti-Rabinowitz type result (see [13, 14]) can be extended to system (1.1) with superlinear and subcritical nonlinear terms by imposing the Ambrosetti-Rabinowitz conditions and other proper conditions on nonlinear terms. Now, we have a natural question of when system (1.1) has multiple nonnegative solutions. In order to obtain the multiplicity of nonnegative solutions for system (1.1), we need only to construct the multiple critical values of the corresponding energy functional. For this matter, first we get two ground states in view of the Nehari procedure, establish a linking structure by the abstract linking theorem on the product space, and then construct the third critical value by the classical minimax principle.
The present paper is organized as following. In Section 2, we provide a linking structure for our variational problem (see Theorem 2.2); in Section 3, we verify that the energy functional satisfies the (P.S.) condition (see Theorem 3.1); in Section 4, we prove the existence of three nonnegative solutions for system (1.1) as λ is small enough, and, in addition, one of them is positive if the equations excluding coupled terms have both a unique positive solution (see Theorem 4.1).
2 Linking structure
In this section, we list some preliminaries, including the abstract linking theorem on the product space and the concept of Nehari manifold, and then we give a linking structure, which is useful for constructing the critical value of the functional associated with system (1.1).
Definition 2.1 
Let f be a real functional on Banach space X and c be a real constant, we say that has the sphere property, if the following hypotheses are satisfied:
(S1) f is continuous on X;
(S2) there is a homeomorphic mapping between and unit sphere of X;
(S3) for any fixed , the equation has a unique solution ;
(S4) X is separated into two open connected subsets by and the origin is contained in one of the subsets.
Theorem 2.1 
Let X, Y be Banach spaces with the following direct sum decomposition:
where , are finite dimensional subspaces of X, Y, respectively. Let f, g be the real functionals on X, Y, respectively, c, d be two real constants and , have the sphere property. Take such that and . Denote
then ∂Q links S.
Definition 2.2 
Assume that is such that , then the constraint set is called a Nehari manifold of X.
Let , and
Denote by the Nehari manifold of on ; we have the following.
Lemma 2.1 Assume that the following conditions are satisfied:
(F1) , and there exist constants and such that
(F2) there exist constants such that
(F3) uniformly w.r.t. , here is the first eigenvalue of the operator on ;
(F4) is an increasing function of u on .
Then the Nehari manifolds , have the sphere property.
Proof It is very similar to the proof of Lemma 4.1 on page 72 in , so we omit it. □
Theorem 2.2 Assume that , satisfy conditions (F1)-(F4). Let and , here . If , then ∂Q links S.
Proof Let , then . Let , then and , and by conditions (F1), (F3) and (F4) it is easy to verify that there exists a unique such that , and for , for . Thus
From Lemma 2.1 and Theorem 2.1, ∂Q links S. □
3 Palais-Smale condition
It is well known that the nontrivial critical points of the functional
correspond to nontrivial solutions of system (1.1). In the following, we show that the functional ψ satisfies the (P.S.) condition.
Theorem 3.1 Assume that , satisfy conditions (F1)-(F2), and , satisfy the conditions:
(H1) there exist constants and such that
(H2) there exist constants , and such that
then the functional ψ satisfies (P.S.) condition.
Proof Let be a sequence such that
we need to prove that has a strongly convergent subsequence.
Claim I. is bounded in .
In fact, from (3.1) there exists a constant such that
and for n large enough
Taking , combining with (3.2) and (3.3), we have
as n large enough, where ’s are positive constants, which implies that is bounded.
Claim II. has a strongly convergent subsequence.
From Claim I, there exists a subsequence, relabel it as , such that
In what follows, we show that
In fact, by (3.1) we get
From conditions (F1), (H1), and the compact embedding theorems (see ), we have
where , are the conjugate numbers of , q, respectively. In combination with (3.5) and (3.6), we obtain
4 Nontrivial solutions
First, we consider the least energy critical point of the functional , which is very useful for the multiplicity of nontrivial solutions to system (1.1).
For that purpose, we define
Similar to , we have the following.
Lemma 4.1 Under conditions (F1)-(F4), , and is a critical value of .
Proof It is very similar to the proof of Theorem 4.2 on page 73 in , so we omit it. □
Lemma 4.2 Let satisfy conditions (F1)-(F4), be of class and satisfy for all with a constant (). Denote
then, on , we have
for all with a constant.
Proof By the assumptions on , we have
for all , then
Let , then
due to , which means that
The proof is complete. □
Theorem 4.1 Assume that , satisfy conditions (F1)-(F4) and that , satisfy conditions (H1)-(H2) and the following condition:
(H3) for a.a. and all ,
Then, there exists a constant such that system (1.1) has at least three nonnegative solutions for any . Furthermore, if the problems
have both a unique positive solution, then for any , system (1.1) has at least three nonnegative solutions in which there is a nontrivial positive solution.
Proof From Lemma 4.1, there exists such that and (), which implies that and are nonnegative solutions of the following system:
Combining with condition (H3), and are nonnegative solutions of system (1.1) with and .
Now, we prove the existence of the third nonnegative solution.
First, we consider , , as follows:
where ’s are positive constants, which implies that there exists an such that
Let (), and
then and , thus ∂Q links S by Theorem 2.2.
Second, we claim that as λ is small enough. In fact, by Lemma 4.1,
which, in connection with (4.2), implies that
for all .
On the other hand, combining with Lemma 4.2, for any we have
here ’s are proper positive constants. Now let , then for all ,
Hence, is inferred from (4.3) and (4.4).
Finally, we define
Therefore, system (1.1) has at least three nonnegative solutions for .
In addition, if the equations
have both a unique positive solution (), the third nonnegative solution of system (1.1) is actually positive. In fact, if one component of the third solution equals zero, its functional value equals or , which is a contradiction. □
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This work is partly supported by NNSF (11101404, 11201204, 11361053) of China and the State Scholarship Fund (201308620021) of China Scholarship Council.
The authors declare that they have no competing interests.
The authors read and approved the final manuscript.