In this paper, the linking theorem and the mountain pass theorem are used to show the existence of nontrivial solutions for the p-Kirchhoff equations without assuming Ambrosetti-Rabinowitz type growth conditions, nontrivial solutions are obtained.
MSC: 35J60, 35J25.
Keywords:linking theorem; mountain pass theorem; nontrivial solutions
In this paper, we consider the nonlocal elliptic problem of the p-Kirchhoff type given by
Recently, the equation
began to attract the attention of several researchers only after Lion  had proposed an abstract framework for this problem. Perera and Zhang  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 sign-changing solutions in .
The study of Kirchhoff-type equations has been extended to the following case involving the p-Laplacian:
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 ) 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 .
In this paper, the weak solutions of (1) are the critical points of the energy functional
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).
Theorem 1 (Linking theorem)
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 (Mountain pass theorem)
2 Main results
In this section, we give our main theorem. Near the origin, we make the following assumptions.
Caratheodory function f satisfies:
The main results of this paper are the following.
3 Proofs of theorems
First, we give several lemmas.
By Lemmas 1 and 2, Φ satisfies the (C) condition. Then the conclusion follows from Theorem 1 and Remark 1. □
Example 1 Set
Example 2 Set
Remark 3 The result of Theorem 1.1 in  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 .
Example 3 Set
Then similar to , it is easy to verify that satisfies (), (), , with .
The authors declare that they have no competing interests.
CL and JW obtained the results in a joint research. All the authors read and approved the final manuscript.
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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