Fast and slow decaying solutions for $H^{1}$-supercritical quasilinear Schrödinger equations

Abstract

We consider the following quasilinear Schrödinger equations of the form

$$\begin{aligned} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \text{ in }\ \mathbb {R}^N\ \text{ and }\ \underset{|x|\rightarrow \infty }{\lim } u(x)=0, \end{aligned}$$

where $N\ge 3,$ $p>\frac{N+2}{N-2},$ $\varepsilon >0$ and V(x) is a positive function. By imposing appropriate conditions on V(x),  we prove that, for $\varepsilon =1,$ the existence of infinity many positive solutions with slow decaying $O(|x|^{-\frac{2}{p-1}})$ at infinity if $p>\frac{N+2}{N-2}$ and, for $\varepsilon$ sufficiently small, a positive solution with fast decaying $O(|x|^{2-N})$ if $\frac{N+2}{N-2}<p<\frac{3N+2}{N-2}.$ The proofs are based on perturbative approach. To this aim, we also analyze the structure of positive solutions for the zero mass problem.