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November/December 2021 On an Anisotropic $p$-Laplace equation with variable singular exponent
Kaushik Bal, Prashanta Garain, Tuhina Mukherjee
Adv. Differential Equations 26(11/12): 535-562 (November/December 2021).

Abstract

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P) \begin{cases} -\Delta_{H,p}u =\frac{\lambda f(x)}{u^{q(x)}}+g(u) \ \text{ in }\Omega,\\ u > 0 \text{ in }\Omega,\ \ \ \ u=0\text{ on }\partial\Omega, \end{cases} \end{equation*} under the assumption $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\lambda>0$ and $0 < q \in C(\bar \Omega)$. For the purely singular case that is $g\equiv 0$, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$ and $g(u)=u^r$ for $r\in (p-1,p^*-1)$.

Citation

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Kaushik Bal. Prashanta Garain. Tuhina Mukherjee. "On an Anisotropic $p$-Laplace equation with variable singular exponent." Adv. Differential Equations 26 (11/12) 535 - 562, November/December 2021.

Information

Published: November/December 2021
First available in Project Euclid: 4 November 2021

Subjects:
Primary: 35D30, 35J62, 35J75

Rights: Copyright © 2021 Khayyam Publishing, Inc.

JOURNAL ARTICLE
28 PAGES

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Vol.26 • No. 11/12 • November/December 2021
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