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

Abstract

<abstract><p>In this paper, we consider the solutions of the boundary blow-up problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \begin{cases} \Delta u = \frac{1}{u^\gamma} +f(u) \ \ \ \ \mathrm{in}\ \ \ \Omega,\\ \ u&gt;0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{in}\ \ \ \Omega, \\ \ u = +\infty \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{on} \ \ \partial\Omega, \end{cases} \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ \gamma &gt; 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.</p></abstract>