$G$-invariant positive solutions for a quasilinear Schrödinger equation

Abstract

We are concerned with a quasilinear elliptic equation of the form \begin{equation*} -\Delta u+a(x) u-\Delta(|u|^\alpha)|u|^{\alpha-2}u=h(u)\quad \hbox{in }\mathbb R^N, \end{equation*} where $\alpha> 1$ and $N\geq 1$. By using variational approaches, we prove the existence of at least one positive solution of the above equation under suitable conditions on $a(x)$ and $h$. In particular, we are interested in the situation that $a(x)$ is invariant under the finite group action $G$.