Solutions for a class of quasilinear Schr\"{o}dinger equations with critical exponents term

TL;DR

This paper investigates a class of quasilinear Schrödinger equations with critical exponent terms, establishing the existence of positive solutions that exhibit specific decay properties using variational methods.

Contribution

The paper introduces a novel variational approach combined with a change of variables to prove the existence of positive solutions with decay properties for a class of quasilinear Schrödinger equations with critical exponents.

Findings

Existence of positive solutions with local maxima

Solutions decay exponentially at infinity

Method applicable to equations with critical exponents

Abstract

In this paper, we study a class of quasilinear Schr\"{o}dinger equation of the form $$-\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u &=&\lambda|u|^{q-2}u+|u|^{2^*(2\alpha)-2}u,\quad\mbox{in}{\mathbb{R}}^N, $$ where $\varepsilon>0,\lambda>0,q\geq2,\alpha>1/2$ are constants, $N\geq3$. By using change of variable and variational approach, the existence of positive solution which has a local maximum point and decays exponentially is obtained.