Singular limits solution for two-dimensional elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights

Abstract

Given Ω bounded open regular set of \({\mathbb{R}^2}\), \({q_1,\ldots, q_K \in \Omega}\), \({\varrho : \Omega \longrightarrow [0,+\infty)}\) a regular bounded function and \({V: \Omega \longrightarrow [0,+\infty)}\) a bounded fuction. We give a sufficient condition for the model problem $$(P):\qquad-{\Delta}u -{\lambda}\varrho(x)|{\nabla}u|^2 = \varepsilon^{2}V(x)e^u$$ to have a positive weak solution in Ω with u = 0 on ∂Ω, which is singular at each q i as the parameters \({\varepsilon}\) and λ tend to 0, essentially when the set of concentration points q i and the set of zeros of V are not necessarily disjoint.