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

TL;DR

This paper investigates solutions to unbalanced fractional elliptic equations with exponential nonlinearities, establishing existence results in subcritical and critical cases using advanced variational and symmetrization techniques.

Contribution

It introduces new existence results for fractional elliptic problems with exponential nonlinearities, combining Moser-Trudinger inequalities and symmetrization methods.

Findings

Existence of nonnegative solutions in subcritical cases.

Existence of solutions in critical exponential growth scenarios.

Application of fractional Sobolev space inequalities.

Abstract

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.