Dimitri Mugnai

Solitary wavesfor the nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwellequations

In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrodinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

Non-Existence Results for the Coupled Klein-Gordon-Maxwell Equations

Abstract In this paper we obtain some non-existence results for the Klein-Gordon equation coupled with the electrostatic field. The method relies on the deduction of some suitable Pohožaev identity which provides necessary conditions to get existence of nontrivial solutions. The case of Maxwell-Schrödinger type coupled equations is also considered.

Wang’s multiplicity result for superlinear (,)–equations without the Ambrosetti–Rabinowitz condition

We consider a nonlinear elliptic equation driven by the sum of a p– Laplacian and a q–Laplacian where 1 < q ≤ 2 ≤ p < ∞ with a (p − 1)– superlinear Caratheodory reaction term which doesn’t satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the problem has at leat three nontrivial solutions; among them one is positive and one is negative.

The Schrödinger–Poisson System with Positive Potential

We study the existence of radially symmetric solitary waves for a nonlinear Schrödinger-Poisson system. In contrast to all previous results, we consider the presence of a positive potential, of interest in physical applications.

Coupled Klein—Gordon and Born—Infeld-type equations: looking for solitary waves

The existence of infinitely many non–trivial radially symmetric solitary waves for the nonlinear Klein—Gordon equation, coupled with a Born—Infeld–type equation, is established under general assumptions.

Carleman estimates and observability inequalities for parabolic equations with interior degeneracy

Abstract. We consider a parabolic problem with degeneracy in the interior of the spatial domain, and we focus on Carleman estimates for the associated adjoint problem. The novelty of interior degeneracy does not let us adapt previous Carleman estimates to our situation. As an application, observability inequalities are established.

Pseudorelativistic Hartree Equation with General Nonlinearity: Existence, Non-existence and Variational Identities

Abstract We prove several existence and non existence results of solitary waves for a class of nonlinear pseudo-relativistic Hartree equations with general nonlinearities. We use variational methods and some new variational identities involving the half Laplacian.

Existence and multiplicity results for the fractional Laplacian in bounded domains

Abstract In this paper, first we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sublevels of the associated functional, which permits to apply a critical point theorem of mixed nature due to Marino and Saccon.

Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients

Abstract We establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non-smooth coefficients, thus preventing the use of standard calculations in this framework.

Stability of Solutions for Some Classes of Nonlinear Damped Wave Equations

We consider two classes of semilinear wave equations with nonnegative damping which may be of type “on-off” or integrally positive. In both cases we give a sufficient condition for the asymptotic stability of the solutions. In the case of integrally positive damping we show that such a condition is also necessary.