Existence of Solution for a Class of Variational Inequality in Whole $${\mathbb {R}}^N$$ with Critical Growth: The Local Mountain Pass Case

In this paper, we study the existence of solution for a class variational inequality in whole $${\mathbb {R}}^N$$ where nonlinearity has critical growth $$N \ge 2.$$ By combining penalization scheme found del Pino and Felmer (Calc Var 4:121–137, 1996) with method due to Bensoussan Lions (Applications des inéquations variationelles en contrôle stochastique. Dunod, Paris, 1978), improve recent result by Alves et al. (J Math Anal Appl 494:124672, 2021).