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: (-\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 \quad \text{in } \mathbb{R}^{N}, 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.

Topics & Concepts