Marta Calanchi

ELLIPTIC EQUATIONS IN ℝ2 WITH ONE-SIDED EXPONENTIAL GROWTH

We consider elliptic equations in bounded domains Ω⊂ℝ2 with nonlinearities which have exponential growth at +∞ (subcritical and critical growth, respectively) and linear growth λ at -∞, with λ>λ1, the first eigen value of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical groups and Morse index of these solutions are given. Also the case λ

Complex dynamics in a class of reversible equations

We consider the pendulum type equation

Fibers and global geometry of functions

\begin{array}{*{20}{c}} {\ddot u + W\prime (t,u) = h(t)}&{t \in \mathbb{R}{\text{,u}} \in \mathbb{R}{\text{,}}} \end{array}

Solutions of logarithmic type for elliptic and hypoelliptic equations

(E) where the functions W andhsatisfy H1 W∈C2(1[R x R;R;)is 1-periodic in t and T-periodic in u H2 h ∈C(R; R) is 1-periodic and \(\smallint _0^1h(t)dt = 0\) H3 V(tu) = W(t, u) -h(t)uis even int.

Radial and non radial solutions for Hardy-Hénon type elliptic systems

We discuss some extensions of the Trudinger–Moser inequality in a special case of weighted Sobolev spaces

Elliptic equations with one-sided critical growth.

Since the seminal work of Ambrosetti and Prodi, the study of global folds was enriched by geometric concepts and extensions accomodating new examples. We present the advantages of considering fibers, a construction dating to Berger and Podolak’s view of the original theorem. A description of folds in terms of properties of fibers gives new perspective to the usual hypotheses in the subject. The text is intended as a guide, outlining arguments and stating results which will be detailed elsewhere.

Multiple solutions for a Hénon-like equation on the annulus

Consider the polynomial ordinary differential equation of first order where al…, anandf:[0, 1] — IR are continuous functions. We will say that a solutionu(t)of (1) is a closed solution if it is defined in the interval [0, 1] and u(0) = u(1).

Homoclinic solutions to periodic motions in a class of reversible equations

LetP be an elliptic or hypoelliptic linear partial operator. We study the followingL1-BMO regularity problem: doesPu=f∈L1 implyu∈BMO? The related problem of the existence forP of fundamental solutions of logarithmic type is also discussed.SuntoSiaP un operatore lineare alle derivate parziali ellittico o ipoellittico. Viene studiato il seguente problema di regolaritàL1-BMO: Pu=f∈L1 implicau∈BMO? Viene altresì studiato il problema connesso dell’esistenza perP di soluzioni fondamentali di tipo logaritmico.