Ground State Solution To Kirchhoff Problem With Sign-changing Potential And Logarithmic Nonlinearity

INohimear partial differential equation usually derives irom natural science and engi-neering field,and has a broad range of background.No matter in theory or in practical application,it is of great significance and value.Because it can well explain some impor-tant phenomenon of nature and effectively solve the nonlinear problems,a large number of science researchers have been widely focused on the international dynamic.The nontrivial solution and ground state solution of Kirchhoff equation,the most fundamental and most important nonlinear equation of nonlinear partial differential equation,have drawn extensive attention over the past decades.In this paper,the solution to Kirchhoff problem is discussed by logarithmic Sobolev inequality,mountain pass theorem and monotonicity trick and so on.The thesis consists of two sections.Chapter 1 is the preface.In Chapter 2,we first study the following Kirchhofi problem with a power function nonlinearity and prove the existence of nontrivial solution where ?(?)R3 is a bounded domain with a smooth boundary,a,b>0,p ?(4,6)and V may change sign.We assume that(V1)V ? L3/2(?)and VO=imfx??V(x)>?;(v2)|V-|3/2<aS.The main conclusions are as follows:Theorem 2.1.1.Suppose that(V1)holds,then the problem(0.3)has at least a nontrivial solution.Theorem 2.1.2.Suppose that(V2)holds,then the problem(0.3)has at least a nontrivial solution.Inspired by theorem 2.1.1 and theorem 2.1.2,we can't help wondering whether the power function can be generalized to the general nonlinearity?Aiming at the problem,we consider the following Kirchhoff problem with the general nonlinearity and obtain the ground state solution by truncating technique where ?(?)R3 is a bounded domain with a smooth boundary,a,b>0.We assume that the potential V and the general nonlinearity f satisfy the following conditions:(V3)V-?L3/2(?)(f1)f?C(?×R)and there exist C>0,q?(2,2*),such that|f(x,t)|?C(1+|t|q-1),(x,t)??×R;(f2)limsupt?0 f(x,t)/t=f0 uniformly in x??;(f3)There exist ?>4,R>0,such that 0<?F(x,t)?tf(x,t),x??,|t|?R,where F(x,t)=?0t f(x,s)ds,(x,t)??×R.Note that(V3)is weaker than(V1)and(V2),but we still have the following conclusion.Theorem 2.1.3.Suppose that(V3)and(f1)-(f3)hold,then the problem(0.4)has a ground state solution.