- Open Access
Existence of nontrivial solutions for p-Kirchhoff type equations
© Liu et al.; licensee Springer. 2013
- Received: 13 July 2013
- Accepted: 29 November 2013
- Published: 30 December 2013
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.
- linking theorem
- mountain pass theorem
- nontrivial solutions
where is a bounded domain, and is the p-Laplacian with .
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 .
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.
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 .
When , we can take , the second eigenvalue of −△ in .
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 
possesses a convergent subsequence; Φ satisfies the (C) condition if Φ satisfies for all .
Definition 2 
A subset A of E is link (with respect to Φ ) to B of E if , for every , there is such that .
Theorem 1  (Linking theorem)
then c is a critical value of Φ.
Remark 1 If Φ satisfies the (C) condition, then Theorem 1 still holds.
Theorem 2  (Mountain pass theorem)
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:
() uniformly in .
uniformly in .
() uniformly in .
uniformly in .
() , .
The main results of this paper are the following.
Theorem 3 Assume that (), () and ()-() hold, then problem (1) has at least one nontrivial weak solution in .
Theorem 4 Assume that (), () and (), (), , hold, then problem (1) has at least one nontrivial weak solution in .
First, we give several lemmas.
Lemma 1 
Under assumptions () and (), any bounded sequence such that in as has a convergent subsequence.
Lemma 2 Under assumptions () and (), the functional satisfies the (C) condition.
This is a contradiction. Then is bounded in . By Lemma 1, we see that has a convergent subsequence in . □
Then there exists such that .
Let , and , .
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. □
- (ii)If , as uniformly in , then
where , . Then it is easy to verify that satisfies ()-() with . When , we can use odd expansion to .
, , . Then it is easy to verify that satisfies ()-() with . When , we can use odd expansion to .
Then there exists such that .
Hence there exists , such that .
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  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 .
where p is even, .
Then similar to , it is easy to verify that satisfies (), (), , with .
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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