Existence of positive solutions for a quasilinear Schrödinger equation

This paper is concerned with the quasilinear Schrödinger equation (0.1)−Δu+V(x)u−Δ[(1+u2)12]u2(1+u2)12=μh(u)inRN,where N≥3 and μ>0 is a parameter. Under some appropriate assumptions on the potential V and the nonlinear term h, we establish the existence of a positive solution for (0.1) with sufficiently large μ. Our method is based on a change of variables, monotonicity trick developed by Jeanjean and a priori estimate.