The Research Of Multiple Of Positive Solutions For Some Nonlinear Elliptic Equations

In this paper, we mainly considered the existence of multiple solutions for some elliptic equations with nonlinear boundary condition, and fourth-order el-liptic equations involving sign-changing weight functions, and a class of nonlinear Schrodinger-Poisson system.This thesis consists of four chapters.In Chapter1, we summarize the background of the related problems and state the main results of the present thesis.In Chapter2, we study the existence and multiplicity of positive solutions to the following semilinear elliptic equations with the nonlinear boundary condition Here Ω is a bounded smooth domain in RN (N≥3), and (?)v denotes the derivative along the outer normal of boundary, and λ>0is a parameter. Using the Nehari manifold and concentration-compactness principle, we establish the existence of multiple positive solutions for the p, q with the following cases1<q<2<p<2*; p=2*,1<q<2; q=2b*,2<p<2b*; q=2b*, p=2*; q=2,1<p<2, where2*=2N/N-2,2b*=2(N-1)/N-2.Our main results extend the results of Brown and Zhang given in [20].In Chapter3, we prove the existence of at least two positive solutions to the following fourth-order elliptic equation involving sign-changing weight functions by using Nehari manifold methods. Here Ω is a bounded smooth domain in RN(N≥5), λ>0is a parameter and a, b:Ωâ†'R are smooth functions which may change sign on Ω. We generalize the results in [87] from the case a(x)=b(x)=1to the changing-sign functions, and extend the result of Alves et al. given in [8].In Chapter4, we are concerned with the following nonlinear Schrodinger-Poisson system where fλ(x)=a(x)+λb(x), and λ>0is a parameter, p∈(3,5). Under the suitable assumptions of the functions K, a and b, we obtain the existence of mul-tiple solutions for the Schodinger-Poisson system by using Ljusternik-Schnirlman category principle and energy comparison. Our main results extend the results of Cerami and Vaira given in [26].