Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on RNRN

In this paper, we study the following quasi-linear elliptic equations of the form −∑i,j=1NDj(ai,j(x,u)Diu)+12∑i,j=1NDsai,j(x,u)DiuDju+V(x)u=h(x,u),x∈RN, where h∈C(RN×R,R)h∈C(RN×R,R), N≥3N≥3 and V∈C(RN,R)V∈C(RN,R). Under appropriate assumptions on V(x)V(x) and h(x,u)h(x,u), some existence results for positive solutions, negative solutions and a sequence of high energy solutions are obtained via the perturbation method.