The Study Of Multiple Solutions For Nonlinear Elliptic Equations

This paper is mainly concerned with the existence of multiple solutions for the quasi-linear elliptic problem with natural growth term, the existence of nontrivial solutions forelliptic equations with a general weight and Hardy potential in a new space, and the exis-tence of a positive solution or multiple solutions for the quasilinear Schr¨odinger equationswith critical exponent.In Chapter 2, we consider a class of quasilinear elliptic problems with natural growthterm First, the existence of a positive and a negative minimizer solutions is proved by variationaltheory. Comparing the critical groups at the minimizer solutions and at the critical point0, we get the two minimizer solutions are nonzero. Then, based on the nonsmooth Morsetheory and the properties of exact sequence, we obtain the existence of three nontrivialsolutions without any symmetry assumptions on f(x,u).In Chapter 3, we establish a new Sobolev-Hardy space by a Hardy inequality withgeneral weight, and get a compact imbedding theorem. Next, based on the Morse theory,we discuss the ellipt?ic equations with general weight and Hardy potentialWe compute the critical groups of the functional at zero and at infinity with di?erentbehaviors of the function f(x,u) or its primitive function F(x,u) near zero and nearinfinity, respectively. Then we obtain the existence of nontrivial solutions in Sobolev-Hardy space by comparing the critical groups at zero and at infinity.In Chapter 4, we mainly discuss the second open problem given by J. M. do′o and U.Severo in [1]. we consider a class of quasilinear Schr¨odinger equations involving criticalexponentBy using a change of variable, quasilinear Schr¨odinger equations are reduced to semilinear elliptic equations. Then, Mountain Pass theorem without (PS)c condition in a suitableOrlicz space is employed to prove the existence of a positive standing wave solution.The Chapter 5 mainly discuss the open problem given by Liu Jiaquan etc [2]. Weconsider a class of quasilinear Schr¨odinger equations involving critical exponent??u ? ?(u2)u =αk(x)|u|p?2u +β|u|22??2u, x∈RN.Using a change of variable, quasilinear Schro¨dinger equations are reduced to semilin-ear elliptic equations. By using Lions'second concentration-compactness principle andconcentration-compactness principle at infinity to prove that the (PS) condition holdslocally and by minimax methods and the Krasnoselski genus theory, we establish theexistence of multiple solutions for the quaslinear Schro¨dinger equations with critical ex-ponent.