Existence Of Sign-changing Solutions For Some Nonlinear Elliptic Equations

In this paper,we mainly study the existence of sign-changing solutions for some nonlinear elliptic equations,which include the generalized quasilinear Schrodinger equations,the stationary Kirchhoff problems involving the fractional Laplacian,and the nonlinear Choquard equation involving the fractional Laplacian.The thesis consists of four chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the whole thesis.In Chapter Two,we study the existence of nodal radial solutions for generalized quasilinear Schrodinger equations in RN with critical growth-div(g2(u)?u)+ g(u)g'(u)|?u|2 + V(x)u = K(u),x ? RN,where N?3,g:R?R+ is an even differential function and g'(s)? 0 for all s?0,K:R?R is a continuous function,the potential V(x):RN ?R is positive and radial.We find the critical exponents for a generalized quasilinear Schrodinger equations and obtain the existence of sign-changing solution with k nodes for any given integer k?0.In Chapter Three,we consider the existence of least energy sign-changing so-lutions for a Kirchhoff-type problem involving the fractional Laplacian operatorwhere s?(0,1),N>2s a and b are positive constants,V(x)?C(RN,R)is a positive potential function which satisfies some conditions.By using the constraint variational method and quantitative deformation lemma,we obtain a least energy sign-changing solution ub for the given problem.Moreover,we show that the energy of ub is strictly larger than twice the ground state energy.We also give a convergence property of ub as b(?)0,where b is regarded as a parameter.In Chapter Four,we investigate the existence of nontrivial solutions for nonlin-ear fractional Choquard equation(-?)su+V(x)u=(I?*|u|p)|u|p-2u,x?RN,where s ?(0,1),N>2s,0<?<N,p?(N+?/N,N+?/N-2s),V(x)?C(RN,R)is a positive potential function which satisfies some conditions.I? is a Riesz potential defined at each point x?RN\{0} byWe prove,via constraint variational method,that the problem has an nonnega-tive groundstate solution.Moreover,by using Fountain Theorem,we obtain an unbounded sequence of solutions.When p?(2,N+?/N-2s),combining constraint vari-ational method and quantitative deformation lemma,we prove that the problem possesses one least energy sign-changing solution.