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.
Categories: math.AP