Singular limits for 2-dimensional elliptic problems involving exponential nonlinearities with sub-quadratic convection term (Q2845571)

Consider the following semilinear elliptic equation NEWLINE\[NEWLINE\begin{cases} -\Delta u-\lambda |\nabla u|^q=\rho^2e^u & \text{in} \;\Omega \\ u=0 & \text{on} \;\partial \Omega \end{cases}\tag{1}NEWLINE\]NEWLINE where \(\Omega\) is an open smooth bounded domain in \(\mathbb{R}^2\), \(q\in [1,2)\), and \(\lambda\), \(\rho\) are positive parameters.NEWLINENEWLINEGiven \(x_1,x_2,\dots,x_m\in \Omega\), the paper is mainly concerned with the existence of positive weak solutions \(u_{\rho,\lambda}\) that tends to a singular function at each \(x_i\) as the parameters \(\rho,\lambda\rightarrow 0\). More specifically, define the Green's function \(G(x,x')\) on \(\Omega\times \Omega\) to be the solution of NEWLINE\[NEWLINE\begin{cases} -\Delta G(x,x')=8\pi \delta_{x=x'} & \text{in} \;\Omega \\ G(x,x')=0 & \text{on} \;\partial \Omega \end{cases}NEWLINE\]NEWLINE and let its regular part \(H(x,x')=G(x,x')+4\log|x-x'|\). Then define NEWLINE\[NEWLINEF(x_1,\dots,x_m)=\sum_{j=1}^m H(x_j,x_j)+\sum_{i\neq j}G(x_i,x_j).NEWLINE\]NEWLINE Assume that \((x_1,\dots,x_m)\) is a non-degenerate critical point of \(F\). Using a nonlinear domain decomposition method the authors are able to construct a family of solutions \(u_{\rho,\lambda}\) of (1) such that NEWLINE\[NEWLINE\lim_{\rho, \lambda\rightarrow 0} u_{\rho,\lambda}=\sum_{j=1}^m G(x_j,\cdot)NEWLINE\]NEWLINE in \(C_{\mathrm{loc}}^{2,\alpha}(\Omega-\{x_1,\dots,x_m\}).\)