Continuity of solutions of a nonlinear elliptic equation

Université Aix-Marseille 1, LATP UMR6632 3 place Victor Hugo, 13331 Marseille Cedex 3, France
bousquet@cmi.univ-mrs.fr

Received: 11 March 2011
Revised: 10 October 2011

Abstract

We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when a is radial: $\mathit{a}\mathrm{\left(}\mathit{\xi }\mathrm{\right)}\mathrm{=}\frac{\mathit{l}\mathrm{\left(}\mathrm{\right|}\mathit{\xi }\mathrm{\left|}\mathrm{\right)}}{\mathrm{\left|}\mathit{\xi }\mathrm{\right|}}\mathit{\xi }$ for some increasing l:ℝ⁺ → ℝ⁺.