On Some Nonlinear Variational Elliptic Equations

In this thesis,we studied three kinds of nonlinear elliptic problems through variational method.In Chapter 2,for strongly indefinite problems,we proved a new fountain theorem(see The-orem 2.2 in chapter 2).We removed the ?-upper semi-continuous assumption on the variational functional which was required in existed references.So,our Fountain Theorem can be used to deal with some strongly indefinite elliptic problems with general indefinite nonlinearities.As an application,in this paper we discussed the following elliptic Schrodinger equation:-?u + V(x)u=g(x)|u|q-2u+h(x)|u|p-2u,u?H1(RN),N?2,(P1)where 1<q<p/(p-1)<2<p<2*= 2N/(N-2),N?3,+?,N=2,and both the potential V(x)and theweight g(x)may change sign.Using our fountain theorem(i.e.,Theorem 2.2),we proved that problem(P1)has infinitely many solutions.In Chapter 3,we studied a Schrodinger equation with periodic potential as follows:-?u + V(x)u=|u|q-2u = ?|u|r-2u,u?H1,N?2,(P2)where the potential V(x)is 1-periodic in xi,…,xN,2<r<q<2*.If the low order perturbation of the nonlinearity is sufficiently small,that is,?>0 small enough(see(f1)-(f5)and Remark 3 in chapter 2 for more general nonlinearities),we proved the existence of infinitely many solutions for(P2)-First,we used some new techniques to analyze the structure of the(PS)sequence of the variational functional of(P2).Secondly,because of the difficulties caused by the lack of ?-upper semi-continuity,so we used new methods to prove the deformation lemmas.In Chapter 4,we studied the following eigenvalue problem of p-Laplacian equation by con-strained variational method:-?pu + V(x)|u|p-2u = ?|u|p-2u+a|u|s-2u,x?Rn(P3)where p?(1,n)e s =p+ p2/n + a ? 0 and ? ? R are parameters,the nonnegative potential V(x)is coercive.Using constrained variational method,we proved that there exists a critical parameter a*such that(P3)has a ground state if 0<a<a*and has no ground state if a>a*.Then,through the method of energy estimate,we discussed the asymptotic behavior of the ground states of(P3)when a(?)a*.If the potential is of" polynomial type" we gave an accurate blow-up rate for the ground states of(P3).