Nontrivial Solutions of a Class of Kirchhoff Type Problems via Local Linking Method

and Applied Analysis 3 is achieved by φ 1 > 0. Let


Introduction
This paper is concerned with the existence of nontrivial solutions of the following nonlocal Kirchhoff type problem: where Ω is a smooth bounded domain in   ( = 1, 2), ,  > 0, and (, ): Ω ×  is a continuous real function.Similar nonlocal problems model several biological systems where  describes a process which depends on the average of itself, for example, that of the population density; see [1].
Problem (1) is related to the stationary analogue of the equation proposed by Kirchhoff [2] as an extension of the classical D' Alembert wave equation for free vibrations of elastic strings.Kirchhoff 's model takes into account the changes in length of the string produced by transverse vibrations.Some early classical studies of Kirchhoff equations were those of Bernstein [3] and Pohožaev [4].However, (2) received great attention only after Lions [5] proposed an abstract framework for the problem.Some interesting results can be found in [6] and the references theorem.In recent years, Alves et al. [7] and Ma and Rivera [8] have obtained positive solutions of such problems by variational methods.Later Perera and Zhang [9,10] obtained nontrivial solutions of Kirchhoff type problems with asymptotically 4-linear term via Yang index.In [9] using the invariant sets of descent flow, Perera and Zhang got a positive, a negative, and a signchanging solution under the 4-sublinear case, asymptotically 4-linear case, and 4-superlinear case.In [10], the authors considered the 4-superlinear case: In [11], Mao and Zhang used minimax methods and invariant sets of decent flow to prove 4-superlinear Kirchhoff type problems without the PS condition and got multiple solutions.In [12], the paper studies the problem (1) by means of the Morse theory and local linking.
Remark 2. Most of the results on the existence and multiplicity of solutions of (1) were obtained under the above superlinear condition () with or without the evenness assumption.Roughly speaking the role of () is to ensure the boundedness of all () (or ( * )) sequences for the corresponding functional.However, there are many functions which are superlinear but it is not necessary to satisfy () even if 1 <  ≤ 2. For example, where  > 2. Then it is easy to check that () does not hold even for any  >  − 1 > 1.On the other hand, in order to verify (), it is usually an annoying task to compute the primitive function of  and sometimes it is almost impossible.For example, where  > 0.
The aim of this paper is to deal with superlinear problems which do not satisfy ().( 2 ) is weaker than () used in [10].Without (), it becomes more complicated.We do not know in our situations whether the Palais-Smale sequence is bounded.We replace () (or ( * )) sequences with weaker Cerami sequence or () * -sequence (the definition given in the next section).
Remark 3. Compared with the method of invariant sets of descent flow and Morse theory, Yang index, our method is more simple and direct.

Preliminaries
Let  be a real Banach space with a direct sum decomposition: Consider two sequences of subspace: For every multi-index  = ( 1 ,  2 ) ∈  × , we denote by   the space We can know For every  :  →  we denote by   the functional  restricted to   .
Definition 4 (see [14]).Let  ∈ contains a subsequence which converges to a critical point of .
Then  has at least one nontrivial critical point.
Next we let  :  1 0 (Ω) be the Sobolev space equipped with the inner product and the norm Denote by 0 <  1 <  2 < ⋅ ⋅ ⋅ the distinct Dirichlet eigenvalues of −Δ on Ω, and denote by  1 ,  2 ,  3 , . . . the eigenfunction corresponding to the eigenvalues; then Abstract and Applied Analysis 3 is achieved by  1 > 0. Let Recall that a function  ∈  is called a weak solution of (1) if Weak solutions are the critical points of  1 functional: Then

The Existence of the Solutions
Lemma 7. Let F satisfies ( 1 ); the function  has a local linking with respect to  =  1 ⊕  2 , where On the other hand, by (7), then there are constants  6 > 0 and  2 > 0,  0 > Then the conclusion is obtained.
Proof of Theorem 1.By Lemmas 7-10 and Lemma 6, we complete the proof.