Existence Of Solutions For Some Kinds Of Kirchhoff Equation

Problem of kirchhoff type was ?rst proposed by kirchhoff in [17] to describe the changes in length of the string produced by transverse vibrations in physical phenomenon. Many researchers began to make extensively research in equations of this type, after Lions [18] introduced an abstract framework.When a = 1, b = 0, problem of kirchhoff type turn into Schrodinger equation:-△u + V(x)u = g(x, u), x ∈ RN, that we are well familiar with. By making different assumptions for potential functions and nonlinearity, several classic results for problem of kirchhoff type have been obtained on smooth bounded domain or in the whole space in recent years, for example, in [1][2][5][16][24-28] and their references therein.Enlightened by paper [7] and [25], i am going to study the existence of the signed solutions with a super-linear term for a class of kirchhoff problems and the existence of the multiple solutions for a class of kirchhoff type problems.According to the content, the paper is organized as follows:In chapter 1 we mainly collect some basic de?nitions and facts that will be used in this article.In chapter 2 we consider the existence of the signed solutions with a super-linear term for a class of kirchhoff problems.where constants ? is smooth bounded domain in RN. λ is a positive parameter,a, b > 0, f(x, u) ∈ C( ˉ? × R, R).In chapter 3 we consider the existence of the multiple solutions for a class of kirchhoff type problems.where the ? is smooth bounded domain in RN, a, b > 0, f(x, u) ∈ C( ˉ? × R, R).