A singular Liouville equation on planar domains

We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, −Δu=(logu+f(u))χ{u>0}$-\Delta u =(\log u+f(u))\chi _{\lbrace u>0\rbrace }$ in Ω⊂R2$\Omega \subset \mathbb {R}^{2}$ with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional Iε$I_\epsilon$ corresponding to the perturbed equation −Δu+gε(u)=f(u)$-\Delta u + g_\epsilon (u) = f(u)$, where gε$g_\epsilon$ is well defined at 0 and approximates −logu$ - \log u$. We show that Iε$I_\epsilon$ has a critical point uε$u_\epsilon$ in H01(Ω)$H_0^1(\Omega )$, which converges to a legitimate nontrivial nonnegative solution of the original problem as ε→0$\epsilon \rightarrow 0$. We also investigate the problem with f(u)$f(u)$ replaced by λf(u)$\lambda f(u)$, when the parameter λ>0$\lambda >0$ is sufficiently large.