Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical cases

Abstract

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation:\begin{equation*}(-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\beta} \quad \text{in } \mathbb R^N,\end{equation*}where $1< q< p$, $0< s_2\leq s_1< 1$, $ps_1=N$, $\beta\in[0,N)$, and $V,K\colon \mathbb R^N\to\mathbb R$, $f\colon \mathbb R\to \mathbb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential nonlinearity. By combining a Moser-Trudinger type inequality for fractional Sobolev spaces with Schwarz symmetrization techniques and related variational and topological methods, we prove the existence of nonnegative solutions.