An Infinite Number Of Sign Solutions For A Class Of Kirchhoff Type Fourth-order Elliptic Boundary Value Problems

In this article we study the existence and multiplicity of solutions for a fourth-order elliptic boundary value problems of Kirchhoff type as following: where A2 denotes the biharmonic operator, ? (?)RN is a bounded smooth domain, f:?. × R?R is continuous functions.On the one hand, according to the importance of high-order differential e-quations both in theory and practice, much more attention has been paid to such problems by a number of authors, see [1-13] and references therein. Among these lit-eratures, Bartsch[1] used to deal with the problem of a p-laplacion equation and give the proof of the existence and multiplicity of a p-laplacion equation by employing the variational method and cut-off technique. In [2], some new existence theorems on multiple sign-changing solutions of some nonlinear fourth-order beam equation-s were established by Wu by combining the variational method and the method of the invariant set of decreasing flow. Zhou[3] and Yang[4] both have studied the fourth-order boundary value problem of elliptic equations, but Zhou obtained sign-changing solutions by applying the variational method and the critical point theory. Yang's results were more than Zhou's and some new existence theorems on multiple positive, negative, and sign-changing solutions were got by using the method of the invariant set of decreasing flow and the minimax method.One the other hand, in recent years, the problems of Kirchhoff type were also paid attention by a number of authors, see [14-29] and reference and therein. In [14], using the variational method and cut-off funcitonal, Li obtained that Kirchhoff type problems without compactness conditions have at least one positive solutions and in [16] by the same method, Li proved that the existence of positive solutions to Kirchhoff type problems with zero mass. In [15], Liang also studied Kirchhoff type problems but was different from Li. Liang proved the positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior by using topological degree and variational method. In [18], using the variational method and the mountain pass theorem, wang obtained the existence and multiplicity of positive and negative solutions for fourth-order elliptic problems. In [20], Wang also proved the existence solutions for fourth-order elliptic equations of Kirchhoff type by using mountain pass truncation method.Through the understanding of these articles, Wu fully proved the existence and multiplicity of sign-changing solutions for nonlinear fourth-order beam equations and also gave the proof of positive and negative solutions. This article was informative, but beam equation is one dimension and to some extent, high dimensional equa-tions are more difficulty. Zhou and Yang both studied the sign-changing solutions for fourth-order elliptic equations and according to the difficulties of sign-changing solutions, less authors studied the sign-changing solutions for fourth-order elliptic equations. The positive and negative solutions for fourth-order elliptic equations of Kirchhoff type was studied by Wang and the equations were more difficulty than elliptic equations and Kirchhoff type problems. Wang also established the solutions for fourth-order elliptic equations of Kirchhoff type with a parameter A.Sign-changing solutions should have more complicated properties, such as the times of changing sign. Thus, sign-changing solutions are interesting challenges in mathematics. Owing to the shortcomings of the research of sign-changing solutions in [3], problem of fourth-order elliptic equation in [4] and positive and negative solu-tions for fourth-order elliptic solutions of Kirchhoff type. In the present paper, mo-tivated by the above results, we investigate the positive, negative and sign-changing solutions of the equation which was studied in [18]. By applying the method the invariant set of decreasing flow and the variational method, we establish several mul-tiple solutions theorems. Furthermore, an infinitely many sign changing solutions theorem is obtain.