A symmetry result for the p-laplacian equation via the moving planes method

Abstract

Thsi paper is concerned with a positive solution u of the non–homogeneous p–Laplacian equation in an open, bounded, connected subset Ω of Rn with C2 boundary. We assume that u verifies overdetermined boundary conditions and we prove that of us has only one critial point Ω thenΩ is a ball and u is radially symmetric; to prove this result we use the moving planes method introduced by J.Serrien. Using the same technique we also prove that the result is stable in the following sense: the boundary of Ω tends to the boundary of a sphere as the diameter of the critical set u tends to 0.