A singular Liouville equation on planar domains

Abstract

We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, −Δ𝑢 = (log 𝑢 + 𝑓(𝑢))𝜒 {𝑢>0} in Ω ⊂ 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 𝐼 𝜖 corresponding to the perturbed equation −Δ𝑢 + 𝑔 𝜖 (𝑢) = 𝑓(𝑢), where 𝑔 𝜖 is well defined at 0 and approximates − log 𝑢. We show that 𝐼 𝜖 has a critical point 𝑢 𝜖 in 𝐻 1 0 (Ω), which converges to a legitimate nontrivial nonnegative solution of the original problem as 𝜖 → 0. We also investigate the problem with 𝑓(𝑢) replaced by 𝜆𝑓(𝑢), when the parameter 𝜆 > 0 is sufficiently large.