[Paper Review] Multiplicity and stability of the Pohozaev obstruction for Hardy-Schr\\"odinger equations with boundary singularity

Nassif Ghoussoub, Saikat Mazumdar+1 more|arXiv (Cornell University)|Mar 29, 2019

Nonlinear Partial Differential Equations30 references4 citations

TL;DR

This paper establishes the multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity at 0. Using sharp blow-up analysis on high-energy solutions of subcritical problems, it proves the existence of infinitely many sign-changing solutions when $\gamma < \frac{n^2}{4} - 1$ and the principal curvatures at 0 are non-positive but not all zero. It further shows that positive solutions do not exist under $C^1$-perturbations of $h$ when $\Omega$ is star-shaped and $h$ is small, even beyond classical Pohozaev conditions.

ABSTRACT

Let $\\Omega$ be a smooth bounded domain in $\\mathbb{R}^n$ ($n\\geq 3$) such that $0\\in\\partial \\Omega$. In this memoir, we consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(\\Omega)$ for the borderline Dirichlet problem, $-\\Delta u-\\gamma \\frac{u}{|x|^2}- h(x) u = \\frac{|u|^{{2^\\star(s)}-2}u}{|x|^s}$ in $\\Omega$, where $0&lt;s&lt;2$, ${{2^\\star(s)}}:=\\frac{2(n-s)}{n-2}$, $\\gamma\\in\\mathbb{R}$ 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 $\\gamma&lt;\\frac{n^2}{4}-1$ and the principal curvatures of $\\partial\\Omega$ at $0$ are non-positive but not all of them vanishing, then the above equation has an infinite number of (possibly sign-changing) solutions in ${H_{1,0}^2(\\Omega)}$. This complements results of the first and third authors, who had previously shown that if $\\gamma\\leq \\frac{n^2}{4}-\\frac{1}{4}$ and the mean curvature of $\\partial\\Omega$ at $0$ is negative, then the equation has a positive solution. On the other hand, the sharp blow-up analysis also allows us to prove that if the mean curvature at $0$ is non-zero and if the mass (when defined) does not vanish, then there is a surprising stability under $C^1$-perturbations of the potential $h$ of those regimes where no variational positive solutions exist. In particular, and in sharp contrast with the non-singular case (i.e., when $\\gamma=s=0$), we show 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.

Motivation & Objective

• To investigate the existence, non-existence, and multiplicity of variational solutions for a critical Hardy-Schrödinger equation with boundary singularity at 0.
• To understand the stability of the Pohozaev obstruction under $C^1$-perturbations of the potential $h$ in the equation.
• To extend previous results on least energy solutions by establishing the existence of infinitely many high-energy, possibly sign-changing solutions under geometric and spectral conditions on $\gamma$ and the boundary curvature.
• To develop a sharp blow-up analysis framework for subcritical approximations to handle high-energy solutions and derive precise asymptotic behavior.

Proposed method

• Utilizes sharp blow-up analysis on solutions of subcritical approximations to the critical equation to derive precise asymptotic profiles near the singular point 0.
• Applies scaling lemmas and constructs blow-up scales to exhaust all possible concentration regimes of high-energy solutions.
• Employs strong pointwise estimates and compactness theorems to control the behavior of solutions near the boundary and the origin.
• Uses a localized Pohozaev identity to estimate $L^{2^\star(s)}$ and $L^2$ terms, curvature contributions, and boundary terms under different scaling regimes.
• Establishes the existence and uniqueness of a Green's function with specific asymptotic behavior at 0 and infinity, crucial for analyzing the operator's coercivity.
• Applies Kelvin transforms and coercivity arguments to derive pointwise decay and growth estimates, proving uniqueness and existence of the Green's function $\mathcal{G}_p$.

Experimental results

Research questions

• RQ1 Under what geometric and spectral conditions does the Hardy-Schrödinger equation with boundary singularity admit infinitely many high-energy, sign-changing solutions in $H_{1,0}^2(\Omega)$?
• RQ2 Can the classical Pohozaev obstruction be stabilized under $C^1$-perturbations of the potential $h$, even when the domain is star-shaped and $h$ is small?
• RQ3 What is the precise asymptotic behavior of solutions and Green's functions near the boundary singularity when $\gamma < \frac{n^2}{4} - 1$?
• RQ4 How does the mean curvature and mass at 0 influence the existence or non-existence of positive solutions under small perturbations of $h$?
• RQ5 What role does the difference $\beta_+(\gamma) - \beta_-(\gamma)$ play in determining blow-up rates and compactness of solutions?

Key findings

• When $\gamma < \frac{n^2}{4} - 1$ and the principal curvatures of $\partial\Omega$ at 0 are non-positive but not all zero, the equation admits infinitely many high-energy, possibly sign-changing solutions in $H_{1,0}^2(\Omega)$.
• If $\gamma \leq \frac{n^2}{4} - \frac{1}{4}$ and the mean curvature at 0 is negative, the equation has a positive least energy solution, extending prior results.
• When the mean curvature at 0 is nonzero and the mass is also nonzero, the regime of non-existence of positive solutions is stable under $C^1$-perturbations of $h$, even beyond classical Pohozaev conditions.
• For star-shaped domains and $h$ sufficiently close to 0, no positive variational solutions exist, demonstrating a surprising non-existence result that goes beyond the classical Pohozaev obstruction.
• The Green's function $\mathcal{G}_p$ for the operator $-\Delta - \frac{\gamma}{|x|^2}$ on $\mathbb{R}_-^n$ exists, is unique up to a multiple of $|x_1||x|^{-\beta_-(\gamma)}$, and satisfies precise asymptotic behavior at 0 and infinity.
• The blow-up rates of solutions are sharply characterized by the exponents $\beta_+(\gamma)$ and $\beta_-(\gamma)$, with distinct behaviors when $\beta_+(\gamma) - \beta_-(\gamma) = 1$ versus $\neq 1$, leading to compactness and existence results.