A singular Liouville equation on two-dimensional domains

Abstract

We prove the existence of a solution for an equation where the nonlinearity is singular at zero, namely - Δ u= (- u-β+ f(u)) χ{u>} in Ω ⊂ R2 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 Iϵ corresponding to the ϵ-perturbed equation - Δ u+ gϵ(u) = f(u) , where gϵ tends pointwisely to u-β as ϵ→ 0 +. We show that Iϵ possesses a critical point uϵ in H01(Ω), which converges to a genuine nontrivial nonnegative solution of the original problem as ϵ→ 0. We also address the problem with f(u) replaced by λf(u) , when the parameter λ> 0 is sufficiently large. We give examples.