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Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schrödinger Equations with Boundary Singularity

About this Title

Nassif Ghoussoub, Saikat Mazumdar and Frédéric Robert

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 285, Number 1415
ISBNs: 978-1-4704-6119-5 (print); 978-1-4704-7485-0 (online)
DOI: https://doi.org/10.1090/memo/1415
Published electronically: April 25, 2023
Keywords: nonlinear elliptic equations, blow-up, conformal invariance, Hardy inequality, Sobolev inequality, stability

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Table of Contents

Chapters

• 1. Introduction
• 2. Setting up the blow-up
• 3. Scaling Lemmas
• 4.   Construction and exhaustion of the blow-up scales
• 5. Strong pointwise estimates
• 6. Sharp blow-up rates and the proof of Compactness
• 7. Estimates on the localized Pohozaev identity
• 8. Estimates of the $L^{2^\star (s)}$ and $L^2-$terms in the localized Pohozaev identity
• 9. Estimate of the curvature term in the Pohozaev identity when $\beta _+(\gamma )-\beta _-(\gamma )>1$
• 10. Proof of the sharp blow-up rates
• 11. Proof of multiplicity
• A. The Pohozaev identity
• B. A continuity property of the first eigenvalue of Schrödinger operators
• C. Regularity and the Hardy-Schrödinger operator on $\mathbb {R}^{n}_{-}$
• D. Green’s function for the Hardy-Schrödinger operator with boundary singularity on a bounded domain
• E. Green’s function for the Hardy-Schrödinger operator on $\mathbb {R}_{-}^n$

Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb {R}^n$ ($n\geq 3$) such that $0\in \partial \Omega$. We consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(\Omega )$ for the borderline Dirichlet problem, \begin{equation*} \left \{ \begin {array}{llll} -\Delta u-\gamma \frac {u}{|x|^2}- h(x) u &=& \frac {|u|^{2^\star (s)-2}u}{|x|^s} \ \ &\text {in } \Omega ,\\ \hfill u&=&0 &\text {on }\partial \Omega \setminus \{ 0 \} , \end{array} \right .\eqno {(E)} \end{equation*} where $0<s<2$, ${2^\star (s)}≔\frac {2(n-s)}{n-2}$, $\gamma \in \mathbb {R}$ and $h\in C^0(\overline {\Omega })$. We use sharp blow-up analysis on—possibly high energy—solutions of corresponding subcritical problems to establish, for example, that if $\gamma <\frac {n^2}{4}-1$ and the principal curvatures of $\partial \Omega$ at $0$ are non-positive but not all of them vanishing, then Equation (E) has an infinite number of high energy (possibly sign-changing) solutions in $H_{1,0}^2(\Omega )$. This complements results of the first and third authors, who showed in their 2016 article, Hardy-Singular Boundary Mass and Sobolev-Critical Variational Problems, that if $\gamma \leq \frac {n^2}{4}-\frac {1}{4}$ and the mean curvature of $\partial \Omega$ at $0$ is negative, then $(E)$ has a positive least energy solution.

On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at $0$ is nonzero and the mass, when defined, is also nonzero, then there is a surprising stability of regimes where there are no variational positive solutions under $C^1$-perturbations of the potential $h$. In particular, and in sharp contrast with the non-singular case (i.e., when $\gamma =s=0$), we prove non-existence of such solutions for $(E)$ in any dimension, whenever $\Omega$ is star-shaped and $h$ is close to $0$, which include situations not covered by the classical Pohozaev obstruction.