Qualitative properties of solutions of m-Laplace systems

Abstract

We prove regularity results for the solutions of the equation -Delta(m)u = h(x), such as summability properties of the second derivatives and summability properties of 1/vertical bar Du vertical bar. Analogous results were recently proved by the authors for the equation -Delta(m)u = f (u). These results allow us to extend to the case of systems of m-Laplace equations, some results recently proved by the authors for the case of a single equation. More precisely we consider the problem {-Delta(m1)(u) = f (v) u > 0 in Omega, u = 0 on theta Omega {-Delta(m2)(v) = g(u) v > 0 in Omega, v = 0 on theta Omega and we prove regularity properties of the solutions as well as qualitative properties of the solutions. Moreover we get a geometric characterization of the critical sets Z(u) equivalent to {x is an element of Omega vertical bar Du(x) = 0} and Z(v) equivalent to {x is an element of Omega vertical bar Dv(x) = 0}. In particular we prove that in convex and symmetric domains we have Z(u) = {0} - Z(v), assuming that 0 is the center of symmetry