In this paper we study the Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = H(x, u, Du), where the principal term is a Leray-Lions operator defined on W
We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = g(x, u) + f , where the principal term is a Leray-Lions operator defined on W 1 ,p 0 (Ω). The function g(x, u) satisfies suitable growth assumptions, but no sign hypothesis on it is assumed. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.
The main purpose of this paper is to investigate dynamical systems F : R 2 → R 2 of the form F (x, y) = (f (x, y), x). We assume that f : R 2 → R is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x0 = (x0, y0), such that the orbitis bounded, O(x0) converges provided that the set of fixed point of F is totally disconnected and F does not admit periodic orbits of prime period two. The obtained result is used to show that all aperiodic orbits can be removed from the dynamics of the map H of Hénon. The goal can be achieved by perturbing H so that the perturbed map H1 does not have any periodic point of prime period two. After the paper had been written, Prof. Giovanni Di Lena passed away. Basilio, Davide and Mario are truly devastated by this sudden and unexpected loss. They dedicate the paper to the loving memory of Giovanni.Mathematics Subject Classification (2010). Primary 58F08; 58F20; Secondary 39B12.
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