A singular Liouville equation on two-dimensional domains

We prove the existence of a solution for an equation where the nonlinearity is singular at zero, namely \(-\Delta u =(-u^{-\beta }+f(u))\chi _{\{u>0\}}\) in \(\Omega \subset {\mathbb {R}}^{2}\) with Dirichlet boundary condition. The function f grows exponentially, which can be subcritical or critical with respect to the Trudinger–Moser embedding. We examine the functional \(I_\epsilon \) corresponding to the \(\epsilon \)-perturbed equation \(-\Delta u + g_\epsilon (u) = f(u)\), where \(g_\epsilon \) tends pointwisely to \(u^{-\beta }\) as \(\epsilon \rightarrow 0^+\). We show that \(I_\epsilon \) possesses a critical point \(u_\epsilon \) in \(H_0^1(\Omega )\), which converges to a genuine nontrivial nonnegative solution of the original problem as \(\epsilon \rightarrow 0\). We also address the problem with f(u) replaced by \(\lambda f(u)\), when the parameter \(\lambda >0\) is sufficiently large. We give examples.