Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical cases - Kumar, Deepak; Rădulescu, Vicenţiu D.; Sreenadh, Konijeti (Topological Methods in Nonlinear Analysis, 2022)
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|^\ba} \quad \text{in } \mathbb R^N,\end{equation*}where $1< q< p$, $0< s_2\leq s_1< 1$, $ps_1=N$, $\ba\in[0,N)$, and$V,K\colon \mathbb R^N\to\mathbb R$, $f\colon \mathbb R\to \mathbb R$ are continuous functions satisfying some natural hypotheses. Weare 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-Trudingertype inequality for fractional Sobolev spaces with Schwarz symmetrization techniques and related variational and topological methods, weprove the existence of nonnegative solutions.