[Paper Review] Boundary singularities of semilinear elliptic equations with Leray-Hardy potential

Huyuan Chen, Laurent Véron|arXiv (Cornell University)|Oct 1, 2019

Advanced Mathematical Modeling in Engineering27 references4 citations

TL;DR

This paper investigates the existence and uniqueness of solutions to semilinear elliptic equations with a Leray-Hardy potential $\Delta u + \frac{\mu}{|x|^2}u + g(u) = \nu$ in a bounded domain $\Omega \subset \mathbb{R}^N_+$ with $0 \in \partial\Omega$, where $\mu \geq -\frac{N^2}{4}$, $g$ is continuous and nondecreasing, and $\nu$, $\lambda$ are Radon measures. The key contribution is a capacity-based framework that provides necessary and sufficient conditions for solvability when $g$ is a power function, distinguishing behavior based on whether the measure is concentrated at the boundary singularity $0$.

ABSTRACT

We study existence and uniqueness of solutions of (E 1) --$Δ$u + $μ$ |x| ^{-2} u + g(u) = $ν$ in $Ω$, u = $λ$ on $\partial$$Ω$, where $Ω$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$ $\partial$$Ω$, $μ$ $\ge$ -- N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and $ν$ and $λ$ two Radon measures respectively in $Ω$ and on $\partial$$Ω$. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient conditions for the solvability of problem (E 1).

Motivation & Objective

• To establish existence and uniqueness of solutions to semilinear elliptic equations with a Leray-Hardy potential when the singularity lies on the boundary of the domain.
• To analyze how the nature of the solution depends on whether the measure $\nu$ is concentrated at the singular point $0 \in \partial\Omega$.
• To develop a capacity framework for power-type nonlinearities $g(u) = |u|^{p-1}u$, providing necessary and sufficient conditions for solvability.
• To characterize the asymptotic behavior of solutions near the boundary singularity $0$ in terms of the spectral parameter $\mu$ and the exponent $p$.
• To extend previous results on internal singularities to the boundary case, particularly generalizing results from [15] and [21] to the boundary setting.

Proposed method

• Utilizes a weak formulation of the equation $-\Delta u + \frac{\mu}{|x|^2}u + g(u) = \nu$ in $\Omega$, with $u = \lambda$ on $\partial\Omega$, where $\nu$ and $\lambda$ are Radon measures.
• Applies Kato-type inequalities and energy estimates to analyze the behavior of solutions near the boundary singularity at $x = 0$.
• Introduces a capacitary framework based on Bessel capacities adapted to the Hardy potential, tailored to the boundary case $0 \in \partial\Omega$.
• Employs the Emden-Fowler transformation $v(t, \sigma) = r^{-\frac{2}{p-1}} u(r, \sigma)$ with $t = \ln r$ to reduce the problem to a time-dependent ODE on the sphere, enabling spectral analysis.
• Uses the theory of analytic functionals and Sturmian arguments to study the discrete structure of solutions and their blow-up rates near $0$.
• Applies comparison principles and supersolution methods to establish bounds on solutions, particularly showing $u = o(\phi_\mu)$ as $x \to 0$ when $p > p^*_\mu$.

Experimental results

Research questions

• RQ1 Under what conditions does the equation $-\Delta u + \frac{\mu}{|x|^2}u + g(u) = \nu$ in $\Omega$ with $u = \lambda$ on $\partial\Omega$ admit a solution when $0 \in \partial\Omega$?
• RQ2 How does the solvability of the problem depend on whether the measure $\nu$ is concentrated at the boundary singularity $0$?
• RQ3 What is the precise asymptotic behavior of solutions near $x = 0$ for power-type nonlinearities $g(u) = |u|^{p-1}u$?
• RQ4 What is the critical exponent $p^*_\mu$ that separates subcritical and supercritical behavior in the boundary singular case?
• RQ5 Can a capacity-theoretic framework be constructed to characterize the class of measures $\nu$ for which the problem is solvable?

Key findings

• When $g(u) = |u|^{p-1}u$, the problem is solvable if and only if the measure $\nu$ belongs to a specific Bessel capacity associated with $\mu$, $N$, and $p$, providing a necessary and sufficient condition.
• For $p > p^*_\mu$, solutions satisfy $u(x) = o(\phi_\mu(x))$ as $x \to 0$, where $\phi_\mu(x) = |x|^{-\frac{N-2}{2} + \sqrt{\mu + \frac{N^2}{4}}}$ is the ground state solution of $\mathcal{L}_\mu u = 0$.
• When $p = p^*_\mu$ and $\mu = \mu_1 = -\frac{N^2}{4}$, the solution still satisfies $u = o(\phi_\mu)$ near $0$, and the solution is unique and weak.
• For $p = p^*_\mu$ and $\mu > \mu_1$, the solution satisfies $\mathcal{E}_v \subset \{0\}$, implying $u$ is asymptotically zero relative to $\phi_\mu$.
• The set of solutions to the reduced equation on the sphere is discrete, and the convergence to a single element of this set implies uniqueness of the asymptotic profile.
• When $p = 3$ and $N + 4\sqrt{\mu - \mu_1} > 0$, the only possible odd integer exponent for which a solution exists is $p = 3$, under certain spectral conditions.