Abstract
In this paper, the linking theorem and the mountain pass theorem are used to show the existence of nontrivial solutions for the pKirchhoff equations without assuming AmbrosettiRabinowitz type growth conditions, nontrivial solutions are obtained.
MSC: 35J60, 35J25.
Keywords:
linking theorem; mountain pass theorem; nontrivial solutions1 Introduction
In this paper, we consider the nonlocal elliptic problem of the pKirchhoff type given by
where is a bounded domain, and is the pLaplacian with .
Recently, the equation
began to attract the attention of several researchers only after Lion [1] had proposed an abstract framework for this problem. Perera and Zhang [2] obtained a nontrivial solution of (2) by using the Yang index and critical group. They revisited (2) via invariant sets of decent flow and obtained the existence of a positive solution, a negative, and a signchanging solutions in [3].
The study of Kirchhofftype equations has been extended to the following case involving the pLaplacian:
for details see [46]. One of the authors has done some related work on this field. Liu [7] gave infinite solutions to the following equation via the fountain theorem and the dual fountain theorem:
However, to the best of our knowledge, there have been few papers dealing with equation (1) using the linking theorem and the mountain pass theorem. This paper will make some contribution to this research field.
It is well known (see [8]) that the eigenvalue problem
has the first eigenvalue , which is simple, and has an associated eigenfunction . It is also known that is an isolated point of , the spectrum of , which contains at least an eigenvalue sequence and .
Let
be a Banach space with the norm for . be the onedimensional eigenspace associated with , where . Let , where , there exists such that
When , we can take , the second eigenvalue of −△ in .
In this paper, the weak solutions of (1) are the critical points of the energy functional
where , . Obviously, and for all ,
In this paper we use the following notation: denotes the Lebesgue space with the norm ; denotes the Lebesgue measure of the set ; is the dual pairing of the space and ; → (resp. ⇀) denotes strong (resp. weak) convergence. denote positive constants (possibly different).
Definition 1[9]
Let , we say that Φ satisfies the Cerami condition at the level if any sequence , along with
possesses a convergent subsequence; Φ satisfies the (C) condition if Φ satisfies for all .
Definition 2[9]
A subset A of E is link (with respect to Φ ) to B of E if , for every , there is such that .
Theorem 1[9] (Linking theorem)
Letbe a Banach space with. Let, and letbe such that. Define
If Φ satisfies the (PS) condition with
thencis a critical value of Φ.
Remark 1 If Φ satisfies the (C) condition, then Theorem 1 still holds.
Theorem 2[10] (Mountain pass theorem)
LetXbe a real Banach space, and letsatisfy the (C) condition. Suppose, for some, and, ,
Then Φ has a critical valuecharacterized by
where
2 Main results
In this section, we give our main theorem. Near the origin, we make the following assumptions.
Suppose that is a continuous function satisfying the following conditions:
() there exists a constant such that for all ;
() there exists a constant such that for all and , .
Caratheodory function f satisfies:
() For some , there exists a constant such that
The main results of this paper are the following.
Theorem 3Assume that (), () and ()() hold, then problem (1) has at least one nontrivial weak solution in.
Theorem 4Assume that (), () and (), (), , hold, then problem (1) has at least one nontrivial weak solution in.
3 Proofs of theorems
First, we give several lemmas.
Lemma 1[7]
Under assumptions () and (), any bounded sequencesuch thatinashas a convergent subsequence.
Lemma 2Under assumptions () and (), the functionalsatisfies the (C) condition.
We claim that is bounded in . For this purpose, we can suppose that . By (), there exists such that
For large n, set , (3) and (4) imply that there exists such that
This is a contradiction. Then is bounded in . By Lemma 1, we see that has a convergent subsequence in . □
Proof of Theorem 3 We obtain from assumptions (), () and () that for some small, there exists such that
Taking , using the inequality and the Sobolev inequality , we have
For every , if , and , then . By (), we know that
Since , there exists such that
By () and (), there exists such that , . Let , then we have
Hence, for large enough, we have .
By Lemmas 1 and 2, Φ satisfies the (C) condition. Then the conclusion follows from Theorem 1 and Remark 1. □
Remark 2 (i) There exists such that , , , which implies , ,
Then
Hence () is much weaker than AmbrosettiRabinowitz type growth conditions.
(ii) If , as uniformly in , then
Example 1 Set
where , . Then it is easy to verify that satisfies ()() with . When , we can use odd expansion to .
Example 2 Set
where
, , . Then it is easy to verify that satisfies ()() with . When , we can use odd expansion to .
Proof of Theorem 4 We obtain from assumptions (), (), that for some small, there exists such that
Taking , using the inequality and the Sobolev inequality , we have
By () and , there exist , , , and such that ,
Therefore, for , let , , we have
Hence there exists , such that .
Then
Summing up Lemma 1 and Lemma 2, satisfies all the conditions of Theorem 2, then the conclusion follows from Theorem 2. □
Remark 3 The result of Theorem 1.1 in [11] corresponds to our results for the case and replaces (). It is easy to see that () is much weaker than , hence the results of Theorems 3 and 4 extend the results of [11].
Example 3 Set
Then similar to [12], it is easy to verify that satisfies (), (), , with .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CL and JW obtained the results in a joint research. All the authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Nature Science Foundation of China (10971179) and the Project of Shandong Province Higher Educational Science and Technology Program (J09LA55, J12LI53).
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