We show existence of signed solutions with positive energy of the problem $\Delta u+u_+^p-u_-^q=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $<q<1<$, $p<(N+2)/(N-2)$ if $N>2$ and the domain $\Omega\subset\mathbb{R}^N$ is bounded and "sufficiently large.'' Our proof is based on the study of the dynamical system associated with the corresponding parabolic problem and it can be easily extended to more general problems. In particular, it does not rely on the uniqueness of the negative solution in contrast to the variational proof in [2] where the authors obtained signed solutions with negative energy.

By proving some estimates for the derivatives of the transition semigroup relative to a suitable stochastic equation, we prove Schauder estimates for a class of second order linear differential operators with unbounded coefficients, both in the drift and in the diffusion term. We also study asymptotic behaviour of the transition semigroup and under suitable dissipative conditions we prove existence and uniqueness of invariant measures and exponentially mixing property, uniform with respect to initial datum.

The differentiability with respect to the delay of the solution of the retarded differential equation (1.1) below will be shown by using elementary arguments. The investigation is extended to the infinite-dimensional case. An application to heat conduction is also given.

We study linear symmetric positive systems under maximal nonnegative boundary conditions. First we consider the noncharacteristic boundary and nonhomogeneous boundary conditions; in this case we give sufficient conditions on the boundary data in order to have $L^2$ and $H^1$ solutions. The inhomogeneous boundary data are treated directly with the advantage of requiring minimal regularity assumptions. Secondly we consider a boundary value problem with boundary matrix not of constant rank. We assume that the boundary is divided in two parts by an embedded manifold which is the intersection of the reference domain and a noncharacteristic hypersurface. The boundary matrix is negative definite on one side of the boundary with respect to the embedded manifold and is positive semi-definite on the other one. Using also the results of the first part, we discuss the existence of regular solutions.

We prove the existence of infinitely many limit periodic solutions for Duffing-like systems of the type $-\ddot{q} = g(t,q)$, where $g$ is limit periodic in time and has a non-degenerate homoclinic solution to the hyperbolic stationary solution $q \equiv 0$. Moreover, we can deal with almost periodic perturbations of the above class of systems. In this case we prove the existence of almost periodic solutions and we show that these solutions are quasi-periodic whenever the perturbation is quasi periodic.

We investigate the well-posedness of a nonlinear parabolic system modelling inflammation in a one-dimensional tissue; it describes the evolution of concentrations of bacteria and leukocytes, and includes a chemotactic process. We consider the case of bacterial infection occurring initially only at the tissue surface; the initial data for the concentration of bacteria is thus a Dirac mass at the boundary. Existence and uniqueness of the solution with such an initial data is proved. The main point is the uniqueness result, which is obtained by a duality argument.

We investigate the global structure of positive radial solutions of a semilinear elliptic equation $\Delta u+K(|x|)\gamma (u)=0$, and study the uniqueness of ground state solutions of this equation. Our discussion is based on a Pohozaev-type identity and some detailed investigation for the oscillatory and asymptotic behavior of the solutions and their variational functions.