[Paper Review] Boundary isolated singularities of positive solutions of some non-monotone semilinear elliptic equations

Marie‐Françoise Bidaut‐Véron, Augusto C. Ponce+1 more|arXiv (Cornell University)|Feb 3, 2009

Nonlinear Partial Differential Equations23 references2 citations

TL;DR

This paper investigates boundary isolated singularities in positive solutions of the semilinear elliptic equation $-\Delta u = u^q$ in a smooth domain $\Omega \subset \mathbb{R}^N$ with $0 \in \partial\Omega$, where $u = \zeta$ on $\partial\Omega \setminus \{0\}$ and $\zeta$ is nonnegative and smooth. For $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, it establishes an upper bound $u(x) \leq C |x|^{-\frac{2}{q-1}}$ and computes the precise asymptotic limit of $|x|^{\frac{2}{q-1}} u(x)$ as $x \to 0$, relying on spherical solutions and uniqueness arguments.

ABSTRACT

Given a smooth domain $\Omega\subset\RR^N$ such that $0 \in \partial\Omega$ and given a nonnegative smooth function $\zeta$ on $\partial\Omega$, we study the behavior near 0 of positive solutions of $-\Delta u=u^q$ in $\Omega$ such that $u = \zeta$ on $\partial\Omega\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x o 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.

Motivation & Objective

• To understand the precise asymptotic behavior of positive solutions to $-\Delta u = u^q$ near an isolated boundary singularity at $0 \in \partial\Omega$.
• To determine the sharp upper bound for the growth rate of such solutions as $x \to 0$.
• To compute the exact limit of $|x|^{\frac{2}{q-1}} u(x)$ as $x \to 0$ in the subcritical range $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$.
• To analyze the critical case $q = \frac{N+1}{N-1}$, where the behavior differs significantly from the subcritical regime.
• To establish the existence and uniqueness of solutions to related equations on spherical domains as a key technical tool.

Proposed method

• Reduction of the boundary value problem to a related equation on the unit sphere via radial transformation and blow-up analysis.
• Use of the existence and uniqueness of solutions to the spherical problem to infer properties of the original solution near the boundary singularity.
• Application of comparison principles and maximum principle techniques to derive pointwise bounds on $u(x)$.
• Asymptotic analysis using the transformation $v(r,\theta) = r^{\frac{2}{q-1}} u(r\theta)$ to study the behavior as $r \to 0^+$.
• Derivation of the limit $\lim_{x \to 0} |x|^{\frac{2}{q-1}} u(x)$ by analyzing the limit profile on the sphere.
• Use of the sub- and super-solution method in conjunction with spherical symmetry to control the growth of $u$.

Experimental results

Research questions

• RQ1 What is the precise asymptotic behavior of positive solutions $u$ to $-\Delta u = u^q$ near an isolated boundary singularity at $0 \in \partial\Omega$?
• RQ2 How does the growth rate of $u(x)$ as $x \to 0$ depend on the exponent $q$ in the range $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$?
• RQ3 What is the exact value of the limit $\lim_{x \to 0} |x|^{\frac{2}{q-1}} u(x)$ for $q$ in the specified subcritical interval?
• RQ4 How does the behavior change when $q = \frac{N+1}{N-1}$, the critical threshold?
• RQ5 Can the asymptotic profile near the boundary singularity be characterized via solutions on the sphere?

Key findings

• For $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, the solution satisfies the upper bound $u(x) \leq C |x|^{-\frac{2}{q-1}}$ for some constant $C > 0$.
• The limit $\lim_{x \to 0} |x|^{\frac{2}{q-1}} u(x)$ exists and is finite, providing a sharp characterization of the singularity strength.
• The value of the limit depends on the boundary data $\zeta$ and the geometry of $\Omega$ near $0$, and is determined via the solution of a related equation on the sphere.
• The critical case $q = \frac{N+1}{N-1}$ exhibits different behavior, suggesting a threshold for the existence of such singular solutions.
• The existence and uniqueness of solutions to the spherical problem are essential in proving the asymptotic results in the subcritical range.
• The method establishes a precise link between the boundary singularity and the solution of a nonlinear elliptic equation on the unit sphere.