EXISTENCE OF SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT SINGULARITY

Ambesh Kumar Pandey; Rasmita Kar

doi:10.1216/rmj.2023.53.903

Abstract

In this article, our interest is to prove the existence of positive solutions to the semilinear elliptic problem

$\matrix{ { - {\rm{div}}(A(x)\nabla u) - \lambda {u \over {|x|^2 }}} \hfill & = \hfill & {{{f(x)} \over {u^{\eta (x)} }}\quad {\rm{in\Omega }},} \hfill \cr u \hfill & = \hfill & {0\quad {\rm{on}}\partial {\rm{\Omega }}.} \hfill \cr }$

Here ${\rm{\Omega }} \subset \mathbb{R}^N$ ( $N \ge 3$ ) is a bounded domain with $0 \in {\rm{\Omega }}$ , $0 \le \lambda < ((N - 2) / (2))^2$ and $0 < \eta (x) \in C^1 ({\rm{\bar \Omega }})$ . The function $f > 0$ is in suitable Lebesgue spaces. The given problem is notable due to the presence of the variable exponent $\eta (x)$ , and it has singularities on the boundary of the domain as well as at the origin.

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Ambesh Kumar Pandey. Rasmita Kar. "EXISTENCE OF SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT SINGULARITY." Rocky Mountain J. Math. 53 (3) 903 - 913, June 2023. https://doi.org/10.1216/rmj.2023.53.903

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Received: 25 March 2022; Revised: 20 July 2022; Accepted: 20 July 2022; Published: June 2023

First available in Project Euclid: 21 July 2023

MathSciNet: MR4617920

zbMATH: 07731154

Digital Object Identifier: 10.1216/rmj.2023.53.903

Subjects:

Primary: 35B45 , 35J25 , 35J61 , 35J75

Keywords: a priori estimate , Hardy inequality , singularity , ‎variable exponent

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