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: \begin{equation*} (-\Delta)^{s}_{p}u+(-\Delta)^{s}_{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{*}_{s}-2}u \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $1<p<q<\frac{N}{s}$, $q^{*}_{s}=\frac{Nq}{N-sq}$, $\lambda>0$ 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.