Isolated singularities for elliptic equations with convolution terms in a punctured ball

TL;DR

This paper studies isolated singularities in elliptic equations with convolution terms in punctured balls, establishing conditions for solution regularity and classifying singular solutions based on potential and kernel parameters.

Contribution

It extends classical regularity results to variable potentials and characterizes the existence and behavior of singular solutions with convolution kernels.

Findings

Optimal conditions for solution regularity with radially symmetric potentials

Sharp criteria for existence of singular solutions based on kernel parameters

Classification of solutions in two and three dimensions near the singularity

Abstract

The purpose of this article is two-fold. First, we investigate the inequality $$ -\Delta u+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ 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 $u, \Delta u, V(x)u\in L^1_{loc}(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $$\displaystyle -\Delta u + \lambda V(x) u = (K_{\alpha, \beta} * u^p) u^q \quad\text{in } B_1 \setminus \{0\},$$ where $0\leq V\in \mathcal{C}^{0, \nu}( \overline B_1\setminus\{0\})$, $0<\nu<1$, $\lambda, p, q>0$ and $$K_{\alpha, \beta}(x) = |x|^{-\alpha}\log^{\beta}\frac{2e}{|x|}, \quad\text{where } 0 \leq \alpha < N, \beta \in \mathbb{R}.$$ For $N \geq 3$, we establish sharp conditions on the exponents $\alpha, \beta, p, q$ 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.