Nontrival And Multiple Solutions For Semilinear Elliptic Equations In Unbounded Domain

This dissertation collects the main results obtained by the author during the period when he has applied for the M.D. The contents are as follow:In chapter two, we study the existence of nontrivial solution for the following nonlinear Schrodinger equations :where N ≥ 3, f(s) : R → R is a continuous nonlinear function and V(x) is sign-changing in R~N.We prove the existence of nontrivial solution, by using some results about least energy solution for the limit case of problem (1) and the concentration-compactness principle introduced by Lions. We remark that the nonlinearity here do not satisfy the classical AR conditon or superlinear assumptions, instead we assume f(s) is bounded. We know that this kind of problem is associated with the asymptotically linear case for eigenvalue problem, and we also can treat the problem as the peturbation of an eigenvalue problem considered by Szulkin-Willem.In chapter three, we consider the following semilinear elliptic equation with Hardy critical exponent:by investigating the following eigenvalue problem,...