We consider the Dirichlet problem for positive solutions of the equation -Delta(m)(u) = f (u) in a bounded smooth domain Omega, with f locally Lipschitz continuous, and prove some regularity results for weak C-1(<(Omega)over bar>) solutions. In particular when f (s) > 0 for s > 0 we prove summability properties of 1/\Du\, and Sobolev's and Poincare 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. (C) 2004 Elsevier Inc. All rights reserved.

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

B. Sciunzi
2004

Abstract

We consider the Dirichlet problem for positive solutions of the equation -Delta(m)(u) = f (u) in a bounded smooth domain Omega, with f locally Lipschitz continuous, and prove some regularity results for weak C-1(<(Omega)over bar>) solutions. In particular when f (s) > 0 for s > 0 we prove summability properties of 1/\Du\, and Sobolev's and Poincare 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. (C) 2004 Elsevier Inc. All rights reserved.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/132679
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