## A note on radiality of solutions of $$p$$-Laplacian equation.(English)Zbl 0841.35008

The authors study the problem of radiality of solutions of the following problem: $\text{div}(|\nabla u|^{p- 2} \nabla u)+ f(u)= 0\quad\text{in}\quad \Omega,\qquad u(x)> 0$ in a domain $$\Omega$$ which is a ball or $$\mathbb{R}^N$$. For such an equation the classical radiality results of Gidas, Ni and Nirenberg do not apply, because the elliptic operator degenerates when $$\nabla u= 0$$. The authors prove the following result: any positive solution of this equation, assuming suitable boundary data, is radial if it satisfies the additional condition $$\nabla u(x)\neq 0$$ if $$x\neq 0$$. Some results on solutions with a singularity are also proved.