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2022 Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical cases
Deepak Kumar, Vicenţiu D. Rădulescu, Konijeti Sreenadh
Topol. Methods Nonlinear Anal. 59(1): 277-302 (2022). DOI: 10.12775/TMNA.2021.026

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.

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Deepak Kumar. Vicenţiu D. Rădulescu. Konijeti Sreenadh. "Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical cases." Topol. Methods Nonlinear Anal. 59 (1) 277 - 302, 2022. https://doi.org/10.12775/TMNA.2021.026

Information

Published: 2022
First available in Project Euclid: 4 April 2022

Digital Object Identifier: 10.12775/TMNA.2021.026

Keywords: fractional $(p , Moser-Trudinger inequality , nonlocal operators , q)$-equation , Schwarz symmetrization , singular exponential nonlinearity

Rights: Copyright © 2022 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.59 • No. 1 • 2022
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