A Study On Some Nonlinear Elliptic Equations

This thesis mainly focuses on the existence of solutions to two kinds of nonlinear elliptic partial differential equations arising in recent years. One of them is the Kirch-hoff elliptic equation, the other one is the variable exponent elliptic equation. The thesis is divided into four chapters.In Chapter1, we firstly introduce the background of our problems, then we sum-marize the new development in this research field and the main results in this thesis.In Chapter2Section2.1we mainly use linking and Morse theory to obtain three existence results of non-trivial solutions to the following p-Kirchhoff type elliptic equa-tion in a bounded and smooth domain Ω∈RN with Dirichlet boundary condition in whichâ–³pu=div(|(â–½u|p-2â–½u) is the usual p-Laplace operator, such that1<p<N; M:R+â†'R+is a function with positive bound up and below.In Chapter2Section2.2we employ Fountain Theorem and Dual Fountain The-orem to derive several existence results of non-trivial solutions to the following p-Kirchhoff type elliptic equation in a bounded and smooth domain Ω∈RN with Neu-mann boundary conditions in which a/av is the outer unit normal derivative;â–³pu is p-Laplace operator, such that1<p<N.In Chapter3, we consider the existence of infinite positive solutions to a kind of p(x)-Kirchhoff type equation. Under the framework of variable exponent spaces, by imposing some kind of oscillation condition on the nonlinear term, we obtain a sequence of different positive solutions to the following equation in which Ω∈RN is a bounded smooth domain. Moreover, we prove that the W01,p(x) norm and L∞norm of this solution sequence tend to zero.In Chapter4, we investigate the (p1(x),p2(x))-Laplace operator, the properties of the corresponding integral functional and the existence of weak solutions to the related differential equations. We show that the integral functional admits a derivative of type (S+) which induces a homeomorphism between duality space pairs. As applications of the above results, we gave some solution existence results of the Dirichlet boundary problem of the (p1(x),p2(x))-Laplace equation-â–³p1(x)u-â–³p2(x)u=f(x,u),(?)x∈Ω, u(x)=0,(?)x∈(?)Ω. and the Neumann boundary problem of the (p1(x),p2(x))-Laplace equation in which Ω∈RN is abounded smooth domain andâ–³p(x)u=div(|â–½u|p(x)-2â–½u) is the p(x)-Laplace operator.This thesis contributes the following to the nonlinear elliptic equations’research field:by using Morse theory, we discuss some p-Kirchhoff type equation for its multi-plicity of solutions (Theorem2.1,2.2and2.3); by using Fountain Theorem and Dual Fountain Theorem, we discuss some p-Kirchhoff type equation for its multiplicity of solutions (Theorem2.16and2.17); by imposing some oscillation conditions on the nonlinear term, we give out multiple solutions to a kind of p(x)-Kirchhoff equation (Theorem3.7); for the first time we discuss the property of (p1(x),p2(x))-Laplace op-erator, including some existence results of the corresponding equations.