This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{ equation*} \begin{ array}{ rllll} (-\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} \; \text{ in } \mb R^N, \end{ array} \end{ equation*} \noi where $1< q< p$, $0<s_2\leq s_1<1$, $ps_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ 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 methods, we prove the existence of nonnegative solutions.
Categories: math.AP