Research Article | Open Access
Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type
We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.
This paper is concerned with the multiplicity of solutions to the following elliptic systems of Kirchhoff type involving the nonlocal integrodifferential operators: where () is a bounded domain with smooth boundary and are two continuous functions. () are two continuous functions whose properties will be introduced later. () are the nonlocal operators defined by; here is a function such that there exist and () such that A typical example for is given by (). In this case is the fractional Laplace operator , where is defined by here and (). The fractional Laplacian is a classical linear integrodifferential operator of order which gives the standard Laplacian when (see ).
Denote by the linear space of Lebesgue measurable functions such that where and . The space is endowed with the norm
The space denotes the closure of in . By Lemmas 6 and 7 in , the space is a Hilbert space which can be endowed with the norm defined as Since a.e. in , we have that the integral in (8) and (9) can be extended to all .
Let be the Cartesian product of two Hilbert spaces, which is a reflexive Banach space endowed with the norm
Denote by the eigenvalues of the following nonlocal operator eigenvalue problem: Similarly, denote by the eigenvalues of the following nonlocal operator eigenvalue problem:
We say that is a weak solution of system (1) if, for every , one has
The fractional Laplacian and nonlocal operators of elliptic type arise in both pure mathematical research and concrete applications, since these operators occur in a quite natural way in many different contexts. For an elementary introduction to this topic, see  and the references therein. Recently, some elliptic boundary problems driven by the nonlocal integrodifferential operator have been studied in the works [3–8].
Recently, problems involving Kirchhoff type operators have been studied in many papers; we refer to [9–13] in which the authors have used the variational method and topological method to get the existence of solutions.
In this paper, motivated by the above mentioned works, we will use Morse theory to investigate the multiplicity of solutions of problem (1). To the best of our knowledge, there is no effort being made in the literature to study the existence of solutions for problem (1). This paper will make some contribution to this research field.
In order to establish solutions for problem (1), we make the following assumptions.(H1) () are two continuous functions, and there exist constants , , , such that (H2) and are two continuous functions with the subcritical growth; that is, there exist some positive constants , such that hold, where , .(H3)There exists , and such that , , and , implies (H4), , uniformly for all a.e. .
The main result of this paper is as follows.
Theorem 1. If (H1)–(H4) hold, then the problem (1) has at least two nontrivial weak solutions in .
For each , we define the functional as follows: where
It is easy to check that is a weak solution of problem (1) which is equivalent to being a critical point of the functional .
First let us recall the definition of the local linking which plays an important role in our paper.
Definition 2. Let be a Banach space with a direct sum decomposition . The functional has a local linking at 0 with respect to if there is such that
Lemma 3. Assume that (H1) and (H4) hold; then the functional is coercive in ; that is, as .
Proof. From (H4) and the continuity of the potentials and we have that, for some , there exists a positive constant such that Thus, by the Sobolev inequality  and (H1), for , we obtain as . Hence, we have that is coercive in .
Lemma 4. If (H1), (H2), and (H4) hold, then satisfies the condition.
Proof. Let be a (PS) sequence of ; then must be bounded by Lemma 3. Passing to a subsequence if necessary, there exists such that weakly in . Thus, there exists a strictly decreasing subsequence , , such that
Since the potential satisfies (H2) and by remark in  we have
Combining (23) with (24), we obtain
On the other hand, we have
Adding (25) to (26), we conclude that
which implies . So, .
Similarly, we can obtain that . The uniform convexity of yields that converges strongly to in .
Thanks to the fact that () continuously, we get by Lemma 6 in  and (4) that where . Similarly, for , there exists a constant such that
In the following, set , where with is the corresponding eigenfunction of and with is the corresponding eigenfunction of . Eigenvalues and are as in (11) and (12), respectively. Taking we can easily know that is complementary subspace of . Hence we have the following direct sum:
If , from Proposition 9 in , we get Moreover, if , by Proposition 9 in , we have
Lemma 5. Assume that (H1)–(H3) hold. Then the functional has a local linking at the origin with respect to .
Proof. (i) Let . Since
by (32), we have that, for given , there is some small enough such that
Now on , we have by (H1) and (H3) that, for with ,
(ii) Let . By (33), similar to (34) and (35), we obtain by (H1)–(H3) that, for with , where () are positive constants, , and . Thus, (37) implies that for with is small enough. The proof is complete.
Let be a real Banach space and . Suppose is an isolated critical point of with and is a neighborhood of , containing the unique critical point; the group is called the th critical group of at , where and is the th singular relative homology group with integer coefficients.
Lemma 6 (see ). Let be a Banach space and a -functional satisfying the (P.S) condition. Assume that has a local linking to the decomposition near the origin, where . If is the unique critical point of in , then
3. The Proof of Theorem 1
We say that is a homological nontrivial critical point of if at least one of its critical groups is nontrivial. By , we have the following abstract critical point theorem.
Lemma 7 (see ). Let be a real Banach space and let satisfy the condition and be bounded from below. If has a critical point that is homologically nontrivial and is not the minimizer of , then has at least three critical points.
Lemma 8. Let (H1)–(H3) hold. Then is a critical point of and .
Proof of Theorem 1. By Lemmas 3 and 4, is coercive and satisfies the condition. Hence is bounded below. By Lemma 8, is homologically nontrivial critical point of but not a minimizer. Then the conclusion follows from Lemma 7.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the referees for their valuable and helpful suggestions and comments that improved the paper. This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and the Natural Science Foundation of China (11271364 and 10771212).
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