A Study On Some Complicated Nonlinear Elliptic Problems

This PhD thesis considers the existence and multiplicity of the solutions for some kinds of complicated nonlinear elliptic problems. The complicated nonlinear here means that:with nonlinear boundary condition, the operator is nonlinear, external force term does not satisfies the (AR) condition. The thesis consists of six chapters:Chapter 1 is an introduction to this thesis. It includes the background and all the problems will be studied in this dissertation as well as the main results we have obtained.In Chapter 2, we recall some mathematical preliminaries including maximal principle, some important facts in nonlinear functional analysis and elliptic prob-lems, variational method, which will be used in the later chapters.In Chapter 3, we study a kind of elliptic equation with nonlinear boundary condition on bounded domain in which Ω(?)RN N is a bounded smooth domain, N≥3. In the first part, we discuss the case that if the nonlinear term f, g satisfy some asymptotic linear condition, the multiplicity of nodal solutions. In the second part, we discuss the case that if the nonlinear terms are concave-convex, there exists infinitely many nodal solutions. The method here is invariant set and Ljusternik-Schnirelman type mini-max principle.In Chapter 4, we consider the weak solutions for quasilinear elliptic equtions in Musielak-Orlicz-Sobolev space in which Ω is a bounded smooth domain in RN,â†'n denote the outer normal vector with respect to (?)Ω. By a new result-the compact boundary embedding theorem on the boundary in Musielak-Orlicz-Sobolev space, we can stduy the existence and multiplicity of the weak solution for quasilinear elliptic equation in Musielak-Orlicz-Sobolev space.In Chapter 5, we study the existence of single peak and multi-peak solutions for the nonlinear elliptic equation: in which λ∈R,(?)u/(?)n denote the outer normal vector with respect By Brouwer fixed point theory and Lyapunov-Schmidt reduction method, we get the existence of the single peak and multi-peak solutions.In Chapter 6, we consider the multiplicity of solutions for a kind of Schrodinger equation. Here the nonlinear term f(u) satisfies the universal condition, by adding some conditions to the coefficient a(x), in a subset of Pohozaev manifold, we get the existence of a positive, a negative and a nodal solution.