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.

A famous theorem of Sergei Bernstein says that every entire solution u = u(x), x ∈ R 2 , of the minimal surface equation div Duis an affine function; no conditions being placed on the behavior of the solution u.Bernstein's Theorem continues to hold up to dimension n = 7 while it fails to be true in higher dimensions, in fact if x ∈ R n , with n 8, there exist entire non-affine minimal graphs (Bombieri, De Giorgi and Giusti). Our purpose is to consider an extensive family of quasilinear elliptic-type equations which has the following strong BernsteinLiouville property, that u ≡ 0 for any entire solution u, no conditions whatsoever being placed on the behavior of the solution (outside of appropriate regularity assumptions). In many cases, moreover, no conditions need be placed even on the dimension n. We also study the behavior of solutions when the parameters of the problem do not allow the Bernstein-Liouville property, and give a number of counterexamples showing that the results of the paper are in many cases best possible.

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Abstract. In this short paper we prove that, for 3 ≤ N ≤ 9, the problem −∆u = e u on the entire Euclidean space R N does not admit any solution stable outside a compact set of R N . This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.

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Abstract. Several Liouville-type theorems are presented for stable solutions of the equation − u = f (u) in R N , where f > 0 is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

We prove a weak comparison principle in narrow domains for sub-super solutions to − p u = f (u) in the case 1 < p ≤ 2 and f locally Lipschitz continuous. We exploit it to get the monotonicity of positive solutions to − p u = f (u) in half spaces, in the case 2N +2 N +2 < p ≤ 2 and f positive. Also we use the monotonicity result to deduce some Liouville-type theorems. We then consider a class of sign-changing nonlinearities and prove a monotonicity and a one-dimensional symmetry result, via the same techniques and some general a-priori estimates. Mathematics Subject Classification (2000)