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

We use a Poincaré type formula and level set analysis to detect onedimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the formOur setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in R 2 and R 3 and of the Bernstein problem on the flatness of minimal area graphs in R 3 . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach is also flexible to very degenerate operators: as an application, we prove one-dimensional symmetry for 1-Laplacian type operators.

We consider the Dirichlet problem for positive solutions of the equation -Delta(m)(u) = f (u) in a bounded smooth domain Omega, with f positive and locally Lipschitz continuous. We prove a Harnack type inequality for the solutions of the linearized operator, a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results, together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z equivalent to {x is an element of Omega vertical bar D(u)(x) = 0} = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary u is an element of C-2(Omega \ {0})

We consider quasilinear elliptic equations involving the p-Laplacian and singular nonlinearities. We prove comparison principles and we deduce some uniqueness results.

We prove the existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problemThe technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using of the moving plane method, in a nonlocal setting, on the whole R N and by some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem into a different nonlocal problem in a weighted fractional space.

Abstract. We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian operator and singular nonlinearities.

No abstract

We consider sign changing solutions of the equation − m (u) = |u| p−1 u in possibly unbounded domains or in R N . We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The results hold true for m > 2 and m − 1 < p < p c (N, m). Here p c (N, m) is a new critical exponent, which is infinity in low dimension and is always larger than the classical critical one.