Existence of positive solutions for a critical nonlinear Schrodinger equation with vanishing or coercive potentials

TL;DR

This paper establishes the existence of positive solutions for a nonlinear Schrödinger equation with potentials that vanish or are coercive at infinity, considering critical nonlinearities and specific asymptotic behaviors of the potentials.

Contribution

It proves the existence of positive solutions under conditions involving critical exponents and asymptotic decay of the potentials, extending previous results to more general potential behaviors.

Findings

Positive solutions exist under specified decay conditions.

Results cover potentials vanishing or coercive at infinity.

The work extends known existence results to critical nonlinearities.

Abstract

In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as $|x|\rightarrow\infty$ with $0<a,\mu<+\infty$, $b<2,$ $ b\neq0$, $ 0<\frac{s}{b}<1$ and $p=2(N-2s/b)/(N-2).$