Weak and strong singular solutions of semilinear fractional elliptic equations

TL;DR

This paper investigates the behavior of weak and strong singular solutions to semilinear fractional elliptic equations with Dirac measure sources, revealing convergence and singularity properties depending on the nonlinearity exponent.

Contribution

It characterizes the limits of solutions with Dirac measure sources for fractional elliptic equations, including convergence to infinity or strong singular solutions, depending on the exponent range.

Findings

Solutions with Dirac sources converge to classical solutions away from the singularity.

For certain exponents, solutions blow up uniformly as the source strength increases.

The nature of the singularity depends critically on the nonlinearity exponent p.

Abstract

Let $p\in(0,\frac{N}{N-2\alpha})$, $\alpha\in(0,1)$ and $\Omega\subset \R^N$ be a bounded $C^2$ domain containing $0$. If $\delta_0$ is the Dirac measure at $0$ and $k>0$, we prove that the weakly singular solution $u_k$ of $(E_k)$ $ (-\Delta)^\alpha u+u^p=k\delta_0 $ in $\Omega$ which vanishes in $\Omega^c$, is a classical solution of $(E_*)$ $ (-\Delta)^\alpha u+u^p=0 $ in $\Omega\setminus\{0\}$ with the same outer data. When $\frac{2\alpha}{N-2\alpha}\leq 1+\frac{2\alpha}{N}$, $p\in(0, 1+\frac{2\alpha}{N}]$ we show that the $u_k$ converges to $\infty$ in whole $\Omega$ when $k\to\infty$, while, for $p\in(1+\frac{2\alpha}N,\frac{N}{N-2\alpha})$, the limit of the $u_k$ is a strongly singular solution of $(E_*)$. The same result holds in the case $1+\frac{2\alpha}{N}<\frac{2\alpha}{N-2\alpha}$ excepted if $\frac{2\alpha}{N}