Rocky Mountain Journal of Mathematics

Influence of mean curvature on Mountain-pass solutions for Hardy-Sobolev equations

Hassan Jaber

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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.

Article information

Rocky Mountain J. Math., Volume 49, Number 2 (2019), 505-519.

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

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Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]

mean curvature Mountain-pass solutions Hardy-Sobolev equations


Jaber, Hassan. Influence of mean curvature on Mountain-pass solutions for Hardy-Sobolev equations. Rocky Mountain J. Math. 49 (2019), no. 2, 505--519. doi:10.1216/RMJ-2019-49-2-505.

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