Construction of singular limits for a strongly perturbed four-dimensional Navier problem with exponentially dominated nonlinearity and nonlinear terms

Abstract

Given a bounded open regular set Ω ∈ R 4 , x 1 , x 2 ,. .. , x m ∈ Ω, λ, ρ > 0, γ ∈ (0, 1), and Q λ some nonlinear operator (which will be defined later), we prove that the problem 2 u + Q λ (u) = ρ 4 (e u + e γ u) has a positive weak solution in Ω with u = u = 0 on ∂Ω, which is singular at each x i as the parameters λ and ρ tend to 0.