Positive solutions for a class of singular quasilinear Schr\"{o}dinger equations with critical Sobolev exponent

TL;DR

This paper establishes the existence of positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent using variational methods, notably without requiring a weight function for the subcritical term.

Contribution

It introduces a novel approach to prove positive solutions for singular quasilinear Schrödinger equations at critical growth without using weight functions.

Findings

Existence of positive solutions proven via variational methods.

Solutions obtained at critical Sobolev exponent without weight functions.

The method applies to equations with singular and quasilinear features.

Abstract

In this paper we prove the existence of positive solutions of the following singular quasilinear Schr\"{o}dinger equations at critical growth \begin{eqnarray*} -\Delta u-\lambda c(x)u-\kappa\alpha(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u = |u|^{q-2}u+|u|^{2^*-2}u,\quad u\in{D^{1,2}(\mathbb{R}}^N), \end{eqnarray*} via variational methods, where $\lambda\geq0$, $0<\alpha<1/2$, $2<q<2^*$. It is interesting that we do not need to add a weight function to control $|u|^{q-2}u$.