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

In this paper, we consider the solutions of the boundary blow-up problem

$$\begin{array}{r}\{\begin{array}{l}\mathrm{\Delta }u=\frac{1}{u^{\gamma }}+f(u)\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u>0\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=+\mathrm{\infty }\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$$

where $\gamma >0,\mathrm{\Omega }$ 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.