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Regularity, monotonicity and symmetry of positive solutions of \(m\)-Laplace equations. (English) Zbl 1108.35069
Summary: We consider the Dirichlet problem for positive solutions of the equation \(-\Delta_m u \equiv-\text{div}(|Du|^{m-2}Du)=f(u)\) in a bounded smooth domain \(\Omega\), with \(f\) locally Lipschitz continuous, and prove some regularity results for weak \(C^1(\overline\Omega)\) solutions. In particular when \(f(s)>0\) for \(s>0\) we prove summability properties of \(|Du|^{-1}\), 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 \(1<m<2\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50 Maximum principles in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J70 Degenerate elliptic equations
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