Existence results for some elliptic equations involving the fractional Laplacian operator and critical growth

Abstract

In this work we prove some results of existence and multiplicity of solutions for equations of the type (􀀀 ) u + V (x)u = f(x; u) in RN; where 0 < < 1, N 2 , (􀀀 ) denotes the fractional Laplacian, V : RN ! R is a continuous function that satisfy suitable conditions and f : RN R ! R is a continuous function that may have critical growth in the sense of the Trudinger-Moser inequality or in the sense of the critical Sobolev exponent. In order to obtain our results we use variational methods combined with a version of the Concentration-Compactness Principle due to Lions.