Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

Abstract

We consider the Dirichlet problem for positive solutions of the equation −Δ m ( u)= f( u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak C 1( Ω ̄ ) solutions. In particular when f( s)>0 for s>0 we prove summability properties of 1 |Du| , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight | Du| m−2 . The point of view of considering | Du| m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov–Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f( s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1< m<2.