Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity

Abstract

This paper is concerned with the following quasilinear Schrödinger equations:

$$\{\begin{array}{c}-\mathrm{div}\left(g^2\left(u\right)\nabla u\right)+g\left(u\right)g^{\prime }\left(u\right)\mid \nabla u{\mid }^2+V\left(x\right)u=K\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N\text{,}\hfill \\ u\in {\mathcal{D}}^{1,2}\left({\mathbb{R}}^N\right)\text{,}\hfill \end{array}$$

where $N\ge 3$ and $V$, $K$ are nonnegative continuous functions. Firstly by using a change of variables, the quasilinear equation is reduced to a semilinear one, whose associated functional is still not well defined in ${\mathcal{D}}^{1,2}\left({\mathbb{R}}^N\right)$ because of the potential vanishing at infinity. However, by using a Hardy-type inequality, we can work in the weighted Sobolev space in which the functional is well defined. Using this fact together with the variational methods, we obtain a positive solution.