Soliton solutions for a class of quasilinear Schrödinger equations with a parameter

Combining variational methods with perturbation arguments, we study the existence of nontrivial classical solution for the quasilinear Schrödinger equation−Δu+V(x)u+κ2[Δ|u|2]u=l(u),x∈RN, where V:RN→RV:RN→R and l:R→Rl:R→R are continuous function, κ   is a parameter and N≥3N≥3. This model has been proposed in plasma physics and nonlinear optics. Our main contribution is related to the fact that we are able to deal with the case κ>0κ>0.