Existence and multiplicity of solutions for general quasi-linear elliptic equations with sub-cubic nonlinearities

Abstract

We consider the following quasi-linear Schrödinger equation−∑i,j=1NDj(aij(u)Diu)+12∑i,j=1NDsaij(u)DiuDju+V(x)u=f(u),x∈RN,N≥3, which includes the modified nonlinear Schrödinger equations. Combining a p-Laplacian perturbation argument, we obtain the existence of positive solutions to the problem above and multiple solutions with additional symmetry conditions on aij and f. A new perturbation approach is used to treat the sub-cubic nonlinearity. In particular, for f(u)=|u|p−2u, the case p∈(2,4) is also considered.