Existence of solution for a class of variational inequality in whole $\mathbb{R}^N$ with critical growth: The local Mountain pass case

TL;DR

This paper proves the existence of solutions for a class of variational inequalities in the entire space with nonlinearities exhibiting critical growth, using advanced penalization techniques to extend previous results.

Contribution

It introduces an improved method combining two penalization schemes to establish solution existence for variational inequalities with critical growth in bR^N.

Findings

Established existence of solutions under critical growth conditions.

Extended previous results by combining penalization methods.

Applicable to variational inequalities in unbounded domains.

Abstract

In this paper we study the existence of solution for a class of variational inequality in whole $\mathbb{R}^N$ where the nonlinearity has a critical growth for $N \geq 2$. By combining a penalization scheme found in del Pino and Felmer [18] with a penalization method due to Bensoussan and Lions [9], we improve a recent result by Alves, Barros and Torres [1].