On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities

Masato Hashizume; Chun-Hsiung Hsia; Gyeongha Hwang

doi:10.3934/cpaa.2019016

Article Contents

2019, Volume 18, Issue 1: 301-322. Doi: 10.3934/cpaa.2019016

This issue Previous Article Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term Next Article On the positive semigroups generated by Fleming-Viot type differential operators

On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities

1.
Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka 558-8585 Japan
2.
Department of Mathematics, Institute of Applied Mathematical Sciences, National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan
3.
National Center for Theoretical Sciences, No. 1 Sec. 4 Roosevelt Rd., National Taiwan University, Taipei, 10617, Taiwan

* Corresponding author
* Corresponding author

Received: November 30, 2017

Revised: February 28, 2018

Published: August 2018

The first author is supported by Grant-in-Aid for JSPS Research Fellow (JSPS KAKENHI Grant Number JP16J08945).

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Abstract

Let $N ≥ 3$ and $Ω \subset \mathbb{R}^N$ be a $C^2$ bounded domain. We study the existence of positive solution $u ∈ H^1(Ω)$ of

$\begin{align*}\left\{\begin{array}{l}-\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \tau \frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\text{ in }\Omega\\\frac{\partial u}{\partial \nu} = 0 \text{ on }\partial\Omega,\end{array}\right.\end{align*}$

where $τ = 1$ or $-1$, $0 < s <2$, $2^*(s) = \frac{2(N-s)}{N-2}$ and $x_1, x_2 ∈ \overline{Ω}$ with $x_1 ≠ x_2$. First, we show the existence of positive solutions to the equation provided the positive $λ$ is small enough. In case that one of the singularities locates on the boundary and the mean curvature of the boundary at this singularity is positive, the existence of positive solutions is obtained for any $λ > 0$ and some $s$ depending on $τ$ and $N$. Furthermore, we extend the existence theory of solutions to the equations for the case of the multiple singularities.

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Mathematics Subject Classification: Primary: 35J25; Secondary: 35J61.

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