Symmetry of positive solutions of a $p$-Laplace equation with convex nonlinearites

In this paper, we consider the symmetry properties of the positive solutions of a $p$-Laplacian problem of the form

$$\begin{array}{r}\{\begin{array}{l}-{\mathrm{\Delta }}_{p}u=f(x,u),\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=g(x),\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$$

where $\mathrm{\Omega }$ is an open smooth bounded domain in $R^N,N\ge 2$, and symmetric w.r.t. the hyperplane $T_0^{\nu }(\nu$ is a direction vector in $R^N,|\nu |=1$ $)$, $f$: $\mathrm{\Omega }\times R^{+}\to R^{+}$ is a continuous function of class $C^1$ w.r.t. the second variable, $g\ge 0$ is continuous, and both $f$ and $g$ are symmetric w.r.t. $T_0^{\nu }$, respectively. Introducing some assumptions on nonlinearities, we get that the positive solutions of the problem above are symmetric w.r.t. the direction $\nu$ by a new simple idea even if $\mathrm{\Omega }$ is not convex in the direction $\nu$.