Positive solutions for quasilinear Schrödinger equations with superlinear term

In this paper, we study the following quasilinear Schrödinger equation −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=h(u),x∈RN, where N≥3, V∈C(RN,R+) is a given potential, g is a C1 even function with g′(t)≤0 for all t≥0,g(0)=1,limt→+∞g(t)=a,0<a<1. The nonlinearity h∈C(R,R) satisfies superlinear growth at infinity. We get existence results of the positive solution for the above problem. Moreover, we give two applications of our results, which improves recent results in Alves et al. [Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differ Equ. 2015;259:318–343].