The purpose of this article is two-fold. First, we investigate the inequality \( -\Delta u+V(x) u\ge f\quad \text{ in } B_1\setminus \{0\}\subset \mathbbm {R}^N , N \ge 2, \) where \(f\in L^1_{loc}(B_1)\) . 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 \(u, \Delta u, V(x)u\in L^1_{loc}(B_1)\) . 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 \(0\le V\in \mathcal {C}^{0, \nu }( \overline{B}_1\setminus \{0\})\) , \(0<\nu <1\) , \(\lambda , p, q>0\) and \(K_{\alpha , \beta }(x) = |x|^{-\alpha }\log ^{\beta }\frac{2e}{|x|}, \quad \text {where } 0 \le \alpha < N, \beta \in \mathbbm {R}.\) 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 \(u(x) \simeq |x|^{2-N}\) 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.
