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On a perturbed critical semilinear equation with singularity

Part of: Equations and inequalities involving nonlinear operators Elliptic equations and systems

Published online by Cambridge University Press:  27 February 2025

Sarika Goyal  and
Kunnath Sandeep
Sarika Goyal
Affiliation:
Department of Mathematics, Netaji Subhas university of Technology, Dwarka, Delhi, India (sarika1.iitd@gmail.com)
Kunnath Sandeep*
Affiliation:
TIFR Centre for Applicable Mathematics, Post Bag No. 6503, Sharadanagar, Yelahanka New Town, Bangalore, India (sandeep@tifrbng.res.in) (corresponding author)
*
*Corresponding author.
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Abstract

In this article, we investigate necessary and sufficient conditions on the perturbation ρ for the existence of positive least energy solutions of the critical singular semilinear elliptic equation $ -\Delta u = \frac{|u|^{2^{*}(s)-2}}{|x|^s}u + \rho(u) $ with Dirichlet boundary condition in a bounded smooth domain in $\mathbb R^n$ containing the origin, where $2^*(s)=\frac{2(n-s)}{n-2}$, $0\leq s \lt 2 \lt n$. We show that the almost necessary and sufficient condition obtained for the case s = 0 in [1] differs conceptually when $0 \lt s \lt 2$.

Keywords

blow-up analysiscritical growthlack of compactnessleast energy solutionSobolev–Hardy inequality

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J75: Singular elliptic equations 47J30: Variational methods
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 35
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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