Existence Of Solutions For Kirchhoff Equation With Different Nonlinearities

Nonlinear partial differential equations,an important branch of modern mathematics,are widely used in the natural science and engineering ares.Many problems in mechanics,control engineering,ecology and economic systems can be described by it.Kirchhoff equation is one of the most basic and vital nonlinear partial differential equations,which can be used to describe the lateral vibration of elastic ropes and the population density problem in the ecosystem.The existence and multiplicity of its solutions have always been of great interests to the authors.In this paper,Kirchhoff equation is studied in three parts.In the first part,the existence of the ground state solution of Kirchhoff equation with sign-changing nonlin-earities is considered by using Nehari manifold,and the ground state solution is obtained of Kirchhoff equation with sign-changing nonlinearities.The second part considers the ex-istence of infinite solutions of Kirchhoff equation with convolution nonlocal terms by using Fountain theorem.The third part discusses the existence of nontrivial solution of the first modified problem by using modified function and mountain pass theorem when the nonlin-earity term does not satisfy Ambrosetti-Rabinowitz and growth condition.We use modified function,Miranda theorem and quantitative deformation lemma to prove the existence of sign-changing solutions for the second modified problem when the nonlinearity term has no growth condition.Finally,the existence of nontrivial and sign-changing small solutions for the original problem are proved by Moser iteration technique.The thesis consists of four sections.Chapter 1 is the preface.We introduce the research background of Kirchhoff equation and the related research results for Kirchhoff equation in the world,and compare our results with the existing ones.In Chapter 2,we consider the following Kirchhoff problem with sign-changing nonlin-earities-(a+b ?R3 |(?)u|2)(?)u+V(x)u=f(x,u)-K(x)|u|q-2u,x?R3,(0.1)where a,b>0,and q?[4,6).When the potential function V satisfies the compact embed-ding condition,K satisfies K?L?(R3)and K?0 for a.e.x ?M3,and f satisfies some suitable conditions,we obtain(0.1)has a ground state solution.In Chapter 3,we consider the following Kirchhoff problem with convolution nonlin-earites-(a+b ?RN |(?)V|2)?u+V(x)u=(L?*|u|p)|u|p-2u+f(x,u),x?RN(0.2)where a,b are positive constants.When the potential function V satisfies the compact embedding condition,and f satisfies some suitable conditions,then for any p?[2,N+?/N-2),equation(0.3)has infinitely many solutions with high energy.In Chapter 4,we consider the following Kirchhoff problem without growth conditions-(a+b ?R3 |(?)Vu|2)?u+V(x)u=|u|p-2u+?f(u),x?R3,(0.3)where a,b are positive constants,? is a positive parameter,p ?(4,6).We assume that f?C(R)satisfies the following conditions:(F4)limt?0 f(t)/t=0(F5)limt?+? f(t)/t=+?(F6)f(t)t>0,t?0.(F7)f(t)/|t|3 is an increasing function of t ?\{0}.We also assume that the potential function V satisfies the compact embedding condition.We obtain if(F4)and(F5)hold,then for every ?>0 small enough,equation(0.3)has nontrivial small solutions.If(F4),(F6)and(F7)hold,then for every ?>0 small enough,equation(0.3)has sign-changing small solutions.