On singular solutions for a semilinear elliptic equation (Q1902049)

Let \(\Omega\) be a bounded open set in \(\mathbb{R}^n\) with a smooth boundary \(\Gamma\), \(\Sigma\) be a smooth compact \(m\)-dimensional manifold in \(\Omega\) and \(\alpha(x)> 0\) be in \(C^\infty(\Omega)\). The author considers the problem \[ - \Delta u= u^p+ \delta_\Sigma,\quad 0< u\in C^2(\Omega- \Sigma). \] The author proves different results on the existence of solutions for the above problem and on the behavior of the solutions near \(\Sigma\).