Asymptotic Analysis

In this article, we study the pointwise asymptotic behavior of iterated convolutions on the one-dimensional lattice Z . We generalize the so-called local limit theorem in probability theory to complex-valued sequences. A sharp rate of convergence toward an explicitly computable attractor is proved together with a generalized Gaussian bound for the asymptotic expansion up to any order of the iterated convolution.

Motivated by a wide interest in the current literature, in this paper, we study the existence of weak solutions for a wide class of double-phase Dirichlet problems depending on a positive real parameter. More precisely, under natural assumptions on the nonlinear term, we are able to show that the main problem admits only the trivial solution provided that the parameter is small, and at least two L ∞ -bounded nonnegative weak solutions whenever the parameter is sufficiently large. The main approach, of a pure variational nature, is based on a fine analysis of the geometrical structure of the associated energy functional.