Regularity, monotonicity and symmetry of positive solutions of \(m\)-Laplace equations.

*(English)*Zbl 1108.35069Summary: 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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\textit{L. Damascelli} and \textit{B. Sciunzi}, J. Differ. Equations 206, No. 2, 483--515 (2004; Zbl 1108.35069)

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