We consider the elliptic equation with boundary singularities \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=-\lambda |x|^{-s_{1}}|u|^{p-2}u+|x|^{-s_{2}}|u|^{q-2}u & \text{ in } \varOmega , \\ u(x)=0 & \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}\) where \(0\in \partial \Omega \) , \(\varOmega \subset \mathbb {R}^N \) , \(N\ge 3\) , \(\lambda >0\) , \(0\le s_1< s_2 < 2\) , \(2<q<p< 2^{*}(s_1)\) , \(q\le 2^{*}(s_2)\) . This paper attempts to study the problem, aiming to provide new research approaches for completely addressing the Li–Lin open problem proposed by Y.Y. Li and C.-S. Lin (Arch. Ration. Mech. Anal. 203(3): 943–968, 2012). Specifically, we focus on its subcritical approximations for q in the range \(2^{*}(s_2)>q>\frac{2-s_2}{2-s_1}p+\frac{2s_2-2s_1}{2-s_1}\) . To address the challenge that the energy functional associated with the equation is unbounded below on the Nehari manifold and fails to admit a global minimizer, we establish the existence of a positive solution that serves as a local minimizer of the energy functional on this manifold. Additionally, we investigate the asymptotic behavior of the positive solution and, through blow-up analysis, identify a new class of blow-up points located on the domain boundary. These boundary blow-up points exhibit distinct characteristics from those commonly reported in the literature.
