Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Strict convexity of level sets of solutions of some nonlinear elliptic equations

Francesca Gladialia1 and Massimo Grossia2

a1 Dipartimento di Matematica, Università di Roma ‘La Sapienza’, P. le A. Moro 2, 00185 Roma, Italy (gladiali@mat.uniroma1.it)

a2 Dipartimento di Matematica, Università di Roma ‘La Sapienza’, P. le A. Moro 2, 00185 Roma, Italy (grossi@mat.uniroma1.it)

Abstract

In this paper we study the convexity of the level sets of solutions of the problem

S0308210500003255disp001

where f is a suitable function with subcritical or critical growth. Under some assumptions on the Gauss curvature of ∂Ω, we prove that the level sets of the solution of (0.1) are strictly convex.

(Received July 10 2003)

(Accepted October 07 2003)