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

The purpose of this article is two-fold. First, we investigate the inequality $$-\mathrm{\Delta }u+V(x)u\ge fin{B}_1\setminus \{0\}\subset {\mathbb{R}}^N,N\ge 2,$$ 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 $u,\mathrm{\Delta }u,V(x)u\in L_{loc}^1(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $${\displaystyle -\mathrm{\Delta }u+\lambda V(x)u=(K_{\alpha ,\beta }\ast u^p)u^q\text{in }{B}_1\setminus \{0\},}$$ where $0\le V\in {\mathcal{C}}^{0,\nu }({\bar{B}}_1\setminus \{0\})$, $0<\nu <1$, $\lambda ,p,q>0$ and $$K_{\alpha ,\beta }(x)=\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }{\log }^{\beta }\frac{2e}{\mathrm{\mid }x\mathrm{\mid }},\text{where }0\le \alpha <N,\beta \in \mathbb{R}\mathrm{.}$$ For $N\ge 3$, we establish sharp conditions on the exponents $\alpha ,\beta ,p,q$ under which singular solutions exist and exhibit the asymptotic behavior $u(x)\simeq \mathrm{\mid }x{\mathrm{\mid }}^{2-N}$ 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.