Abstract
In this paper, we deal with the existence of three nontrivial solutions for the potential system of pLaplacian 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.
MSC: 35J20, 35J25.
Keywords:
elliptic equations; Nehari manifold; linking theorem; minimax principle1 Introduction
In this paper, we are concerned with the multiplicity of nontrivial solutions for the following system of quasilinear equations:
where
In recent years, many authors have studied the existence of nontrivial solutions for
Laplacian systems and pLaplacian systems, see [15] 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 [2] Costa and Magalhaes unified the cooperative and noncooperative Laplacian systems,
and they got the existence of nontrivial solutions via the variational approach; in
[1] 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 pLaplacian systems, and they obtained the existence of positive solutions by the method
of lower and upper solutions and LeraySchauder degree theory, respectively; in [5] Zhang and Zhang considered the existence of nontrivial solutions for nonlinear Laplacian
systems and pLaplacian systems applying the direct variational method. More recently, some authors
discussed the multiplicity of nontrivial solutions for Laplacian systems and pLaplacian systems, see [612] 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
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 AmbrosettiRabinowitz type result (see [13,14]) can be extended to system (1.1) with superlinear and subcritical nonlinear terms by imposing the AmbrosettiRabinowitz 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[12]
Let f be a real functional on Banach space X and c be a real constant, we say that
(S_{1}) f is continuous on X;
(S_{2}) there is a homeomorphic mapping between
(S_{3}) for any fixed
(S_{4}) X is separated into two open connected subsets by
Theorem 2.1[12]
LetX, Ybe Banach spaces with the following direct sum decomposition:
where
then∂QlinksS.
Definition 2.2[15]
Assume that
Let
Denote by
Lemma 2.1Assume that the following conditions are satisfied:
(F_{1})
(F_{2}) there exist constants
(F_{3})
(F_{4})
Then the Nehari manifolds
Proof It is very similar to the proof of Lemma 4.1 on page 72 in [15], so we omit it. □
Theorem 2.2Assume that
Proof Let
From Lemma 2.1 and Theorem 2.1, ∂Q links S. □
3 PalaisSmale 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 3.1Assume that
(H_{1}) there exist constants
(H_{2}) there exist constants
then the functionalψsatisfies (P.S.) condition.
Proof Let
we need to prove that
Claim I.
In fact, from (3.1) there exists a constant
and for n large enough
Taking
as n large enough, where
Claim II.
From Claim I, there exists a subsequence, relabel it as
In what follows, we show that
In fact, by (3.1) we get
that is,
From conditions (F_{1}), (H_{1}), and the compact embedding theorems (see [16]), we have
where
By the property (S_{+}) of pLaplacian operator (see [14,17]), we know that
4 Nontrivial solutions
First, we consider the least energy critical point of the functional
For that purpose, we define
where
Similar to [15], we have the following.
Lemma 4.1Under conditions (F_{1})(F_{4}),
Proof It is very similar to the proof of Theorem 4.2 on page 73 in [15], so we omit it. □
Lemma 4.2Let
then, on
for all
Proof By the assumptions on
for all
Let
due to
The proof is complete. □
Theorem 4.1Assume that
(H_{3}) for a.a.
Then, there exists a constant
have both a unique positive solution, then for any
Proof From Lemma 4.1, there exists
Combining with condition (H_{3}),
Now, we prove the existence of the third nonnegative solution.
First, we consider
where
Let
then
Second, we claim that
which, in connection with (4.2), implies that
for all
On the other hand, combining with Lemma 4.2, for any
here
Hence,
Finally, we define
In combination with Theorem 3.1 and the classical minimax principle (see [15,18]),
Therefore, system (1.1) has at least three nonnegative solutions for
In addition, if the equations
have both a unique positive solution (
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Acknowledgements
This work is partly supported by NNSF (11101404, 11201204, 11361053) of China and the State Scholarship Fund (201308620021) of China Scholarship Council.
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