# Fast and slow decaying solutions for $H^{1}$-supercritical quasilinear Schr\"{o}dinger equations

## TL;DR

This paper investigates the existence of multiple positive solutions with different decay rates for a class of supercritical quasilinear Schrödinger equations, using perturbative methods and analysis of the zero mass problem.

## Contribution

It establishes the existence of infinitely many slow-decaying solutions and a fast-decaying solution under specific conditions, advancing understanding of solution behaviors in supercritical regimes.

## Key findings

- Existence of infinitely many slow-decaying solutions for ε=1.
- Existence of a fast-decaying solution for small ε.
- Analysis of the zero mass problem structure.

## Abstract

We consider the following quasilinear Schr\"{o}dinger equations of the form \begin{equation*} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \mbox{in}\ \mathbb{R}^N\ \mbox{and}\ \underset{|x|\rightarrow \infty}{\lim} u(x)=0, \end{equation*} where $N\geq 3,$ $p>\frac{N+2}{N-2},$ $\varepsilon>0$ 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.