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.
Categories: math.AP