Singular limits solution for 2-dimensional elliptic problems involving exponential nonlinearities with sub-quadratic convection nonlinear gradient terms and singular weights

Abstract

Given a bounded open regular set Ω of ℝ2$\mathbb {R}^2$, q1,...,qK∈Ω${q_1, \ldots , q_K \hspace*{-0.85358pt}\in \hspace*{-0.85358pt} \Omega }$, a regular bounded function ϱ:Ω→[0,+∞)${\varrho \hspace*{-0.56905pt}:\hspace*{-0.56905pt} \Omega \hspace*{-0.85358pt}\rightarrow \hspace*{-0.85358pt} [0,+\infty )}$ and a bounded function V:Ω→[0,+∞)${V: \Omega \rightarrow [0,+\infty )}$, we give a sufficient condition for the model problem -Δu-λϱ(x)|∇u|q=ε2V(x)eu$ -\Delta u -\lambda \varrho (x)\vert \nabla u\vert ^q = \varepsilon ^2 V(x) e^u $ to have a positive weak solution in Ω with u = 0 on ∂Ω${\partial \Omega }$, which is singular at each qi as the parameters ε and λ tend to 0, without considering any relation between them, essentially when the set of concentration points qi and the set of zeros of V are not necessarily disjoint and q∈[1,2)$q\in [1,2)$ is a real number.

Keywords

• singular limits
• green's function
• gradient term with sub-quadratic growth
• nonlinear domain decomposition method
• 35j60
• 53c21
• 58j05