Fractional $p\&q$ Laplacian Problems in ${\mathbb{R}}^N$ with Critical Growth

Abstract

We deal with the following nonlinear problem involving fractional $p\&q$ Laplacians:

where $s\in (0,1)$, $1<p<q<\frac{N}{s}$, $q_s^{\ast }=\frac{Nq}{N-sq}$, $\lambda >0$ is a parameter, $h$ is a nontrivial bounded perturbation and $f$ is a superlinear continuous function with subcritical growth. Using suitable variational arguments and concentration-compactness lemma, we prove the existence of a nontrivial non-negative solution for $\lambda$ sufficiently large.