Existence And Multiplicity To Solutions For Nonlinear Schrdinger-type Equations

Nonlinear problems usually arise in the natural science and engineering areas. Because they can well explain the various natural phenomenon, a large number of researchers have paid attention to the problems for a long time. Moreover for Schrodinger equations as the most fundamental in quantum mechanics, the existence, multiplicity and nonexistence of solutions have also been the hot spot in recent years.In this paper, we use variational methods, such as the fountain theorem, the dual fountain theorem and constrained minimization arguments, to discuss the existence and multiplicity of solutions for two special kinds of Schrodinger equations.The thesis consists of three sections.Chapter 1 is the preface.In Chapter 2, we consider the following Schrodinger-Possion system where α>0, K≥0 and K≠0. For g and V, we assume:(g1)g{x,t)=βb{x)|t|p-1t+|t|4t, where β > 0,p∈(0,1),b≥0,b∈Ls(R3)\{0},s∈ (6/(5-p),2/(1-p));(g2)9(x,t)=βb(x)|t|P-1t+μ|t|q-1t,where p∈(0,1),q∈(3,5),b≥0,b∈Ls(R3)\{0}, s∈(6/(5-p),2/(1-p));(V) V ∈C(R3)satisfies V{x)≥V0≥0 for all x∈R3;(V1) For every M>0,m({x∈R3:V{x)≤M})<∞, where m denotes the Lebesgue measure in R3;(V2) There exists M0>0 such that m({x∈R3:V(x)≤M0})<∞.We obtain the following results via the fountain theorem and the dual fountain theorem.Theorem 2.1.1 Suppose that (V), (V1), (g1) are satisfied and V0>0,K∈L∞(R3). Then there exists β0 > 0 such that for β∈G (0,β0) and all α>0, the system (2.1) has infinitely many solutions {{uk,φk)} satisfyingTheorem 2.1.2 Suppose that (V), (V2), (g1) hold and V0 = 0,K∈L2(R3). Then there exists β1> 0 such that for β∈(0,β1) and all α>0, the system (2.1) admits infinitely many solutions {(vk>,φk)} satisfyingTheorem 2.1.3 Suppose that (V), (V1), (g2) are satisfied and V0 > 0,K ∈ L∞(R3). Then for every a > 0, we have the following results:(i) For every μ > 0, β ∈ R1, the system (2.1) has a sequence of positive-energy solutions;(ii) For every μ > 0,β∈ R1, the system (2.1) has a sequence of negative-energy solu-tions.In Chapter 3, we study the following quasilinear Schrodinger equation where N > 3,1< p < q < 22* - 1,2* = 2N/(N - 2) and the potential V satisfies(V3) V € C(RN) is radially symmetric and there exist V\ and V2 > 0 such that V\ < V(x) < V2 for all x ∈RN.For the equation, we first transform it to a semilinear one by a change of variables and then conclude the main result in H (R ) using a constrained minimization argument.Theorem 3.1.1 Let N≥ 3,1 < p < q < 22* - 1. Suppose V satisfies (V3). Then there is a0 > 0 such that for a < a0,β∈R1 exists and the equation (3.1) has a nontrivial solution.