On The Existence Of Multiple Solutions To A Class Of Schr?dinger Equation With Concave And Convex Nonlinearities

In this thesis,by using variational methods,we mainly study the following the existence of the solutions to Schrodinger equations with concave and convex nonlin-earities-?u+V(x)u=?|u|^q-2u+|u|^p-2u,x?RN,where q₀<1<2<p<2^*,q₀=max{1,2N/N+2?},2^*=2N/N-2,N?3;2^*=?,N=1,2.The potential function V:R^N?R is a continuous function and satisfies the following conditions(V₁)infV(x)>0;(V₂)there is a constant ?>0 such that ?₀:=(?)|x|^-2?V(x)>0.Firstly,we give Sobolev embedding result and show that the energy functional cor-responding to the above equation is ? C¹(H_V¹(R^N),R);Secondly,there is concerned with the proof of the existence of a sequence of solutions with positive energy by Compactness Theorem and Symmetric Mountain Pass Lemma;Then there is con-cerned with the proof of the existence of a sequence of solutions with negative energy by Dual Fountain Theorem;Lastly,there is concerned with the proof of the existence of infinite negative energy solutions for equations with concave nonlinear terms near the origin by Clark's Theorem.