Isolated singularities for elliptic equations with convolution terms in a punctured ball
Marius Ghergu, Zhe Yu
TL;DR
This work extends the Brezis–Lions theory to elliptic inequalities with radial potentials in a punctured ball and analyzes a nonlocal Choquard-type equation with a logarithmic kernel. It develops sharp integral estimates for the convolution kernel, studies the fundamental solution of -Δ+μI, and establishes a nonlocal sub- and super-solution framework to construct singular solutions with prescribed asymptotics. In dimensions N≥3, the authors identify sharp thresholds on the exponents α,β,p,q that yield singular solutions behaving like |x|^{2−N} near the origin, while in N=2 they classify existence and boundedness based on V’s local near-origin behavior. The results unify local regularity, nonlocal convolution effects, and singular phenomena in punctured domains, offering a comprehensive view of how radial potentials and nonlocal nonlinearities govern isolated singularities.
Abstract
The purpose of this article is two-fold. First, we investigate the inequality $$ -Δu+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we provide optimal conditions for which any solution $0\leq u\in \mathcal{C}^2(B_1\setminus\{0\})$ of the above inequality satisfies $u, Δu, V(x)u\in L^1_{loc}(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $$\displaystyle -Δu + λV(x) u = (K_{α, β} * u^p) u^q \quad\text{in } B_1 \setminus \{0\},$$ where $0\leq V\in \mathcal{C}^{0, ν}( \overline B_1\setminus\{0\})$, $0<ν<1$, $λ, p, q>0$ and $$K_{α, β}(x) = |x|^{-α}\log^β\frac{2e}{|x|}, \quad\text{where } 0 \leq α< N, β\in \mathbb{R}.$$ For $N \geq 3$, we establish sharp conditions on the exponents $α, β, p, q$ under which singular solutions exist and exhibit the asymptotic behavior $u(x) \simeq |x|^{2-N}$ near the origin. For $N = 2$, we provide a classification of the existence and boundedness of solutions based on the local behavior of the potential $V(x)$ near the origin.