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PERTURBATION THEOREMS FOR FRACTIONAL CRITICAL EQUATIONS ON BOUNDED DOMAINS

Published online by Cambridge University Press:  09 March 2020

AZEB ALGHANEMI
Affiliation:
Department of Mathematics, King Abdulaziz University, P.O. Box 80230, Jeddah, Kingdom of Saudi Arabia e-mail: aalghanemi@kau.edu.sa
HICHEM CHTIOUI
Affiliation:
Department of Mathematics, Sfax University, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia e-mail: Hichem.Chtioui@fss.rnu.tn

Abstract

We consider the fractional critical problem $A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$ in $\unicode[STIX]{x1D6FA},u=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , where $A_{s},s\in (0,1)$ , is the fractional Laplace operator and $K$ is a given function on a bounded domain $\unicode[STIX]{x1D6FA}$ of $\mathbb{R}^{n},n\geq 2$ . This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when $K$ is close to 1.

MSC classification

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by F. Cirstea

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under Grant No. (KEP-PhD-41-130-38). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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PERTURBATION THEOREMS FOR FRACTIONAL CRITICAL EQUATIONS ON BOUNDED DOMAINS
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