Symmetry of large solutions for semilinear elliptic equations in a symmetric convex domain

Abstract

In this paper, we consider the solutions of the boundary blow-up problem $ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{on} \ \ \partial\Omega, \end{cases} \end{eqnarray*} $ where $ \gamma > 0, \ \Omega $ is a bounded convex smooth domain and symmetric w.r.t. a direction. $ f $ is a locally Lipschitz continuous and non-decreasing function. We prove symmetry and monotonicity of solutions of the problem above by the moving planes method. A maximum principle in narrow domains plays an important role in proof of the main result.

Keywords

• semilinear elliptic systems
• moving plane method
• maximum principle
• symmetry of large solutions