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2019 Influence of mean curvature on Mountain-pass solutions for Hardy-Sobolev equations
Hassan Jaber
Rocky Mountain J. Math. 49(2): 505-519 (2019). DOI: 10.1216/RMJ-2019-49-2-505

Abstract

Let $\Omega $ be a smooth open domain in $\mathbb{R} ^n$, $n \geq 3$, with $0 \in \partial \Omega $ and $s \in {]}0,2{[}$. Let $2^\star(s) := {2(n-s)}/({n-2})$ be the critical Hardy-Sobolev exponent. Consider the Dirichlet problem that corresponds to the critical and perturbative Hardy-Sobolev equation \begin{aligned}\begin{cases}-\Delta u = {u^{2^\star(s) -1}}/{|x|^s} + h u^{q-1} \quad &\mbox {in } \Omega ,\\u \equiv 0 &\mbox {on } \partial \Omega, \end{cases}\end{aligned} where $q \in {]} 2,2^\star {[}$ with $2^\star := 2^\star (0)$ and $h \in C^0(\overline {\Omega })$, $h \geq 0$, almost everywhere on $\Omega $. In this article, we investigate the influence of perturbation and mean curvature at a boundary singularity on the existence of a positive Mountain pass solution to the above equation.

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Hassan Jaber. "Influence of mean curvature on Mountain-pass solutions for Hardy-Sobolev equations." Rocky Mountain J. Math. 49 (2) 505 - 519, 2019. https://doi.org/10.1216/RMJ-2019-49-2-505

Information

Received: 2 August 2018; Published: 2019
First available in Project Euclid: 23 June 2019

zbMATH: 07079981
MathSciNet: MR3973237
Digital Object Identifier: 10.1216/RMJ-2019-49-2-505

Subjects:
Primary: 35J60, 58J05

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 2 • 2019
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