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

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 $f\in L_{loc}^1(B_1)$. If $V\ge 0$ is radially symmetric, we provide optimal conditions for which any solution $0\le u\in {\mathcal{C}}^2(B_1\setminus \{0\})$ of the above inequality satisfies $u,\hat{I}”u,V(x)u\in L_{loc}^1(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $$-\hat{I}”u+\hat{I}»V(x)u=(K_{\hat{I}\pm ,{\hat{I}}^2}\ast u^p)u^q\text{in }{B}_1\setminus \{0\},$$ where $0\le V\in {\mathcal{C}}^{0,\hat{I}½}({\overline{B}}_1\setminus \{0\})$, $0<\hat{I}½<1$, $\hat{I}»,p,q>0$ and $$K_{\hat{I}\pm ,{\hat{I}}^2}(x)=|x{|}^{-\hat{I}\pm }{\log }^2\frac{2e}{|x|},\text{where }0\le \hat{I}\pm <N,{\hat{I}}^2\in \mathbb{R}.$$ For $N\ge 3$, we establish sharp conditions on the exponents $\hat{I}\pm ,{\hat{I}}^2,p,q$ 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.