The Existence Of Positive Solutions For The Schrödinger-Kirhoff-type Equation With A Critical Exponent

This paper is devoted to studying the singular Schrcdinger-Kirchhoff-type problem with critical exponent,the main content as follows:In chapter 1,we summarize some background and the latest progresses on the study of Schrodinger-Kirchhoff-type equation.In chapter 2,we mainly introduces some basic knowledge used in this paper.In chapter 3 and chapter 5,we prove the theorem 1.1,that is for the equation(?)where a,b>0 are constants,?(?)R3 is a smooth bounded domain,?>0 is a real parameter,??(0,1)is a constant,and 0<?<??1(? is the first eigenvalue of-?u = ?|x|s-2 u,under Dirichlet boundary condition).Under appropriate assumption on Q and f,we obtain two positive solutions via the variational and perturbation methods.In chapter 4,we study the existence of at least one mountain-pass solution of the following perturbed problem:(?)where a,b>0 are constants,?(?)R3 is a smooth bounded domain,0<?<1,?>0 is a real parameter,?(?)(0,1)is a constant,and 0<?<??1.