Semiclassical limit for Schr\"odinger equations with magnetic field and Hartree-type nonlinearities

The semi-classical regime of standing wave solutions of a Schr\"odinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.Comment: 34 page

New characterizations of Sobolev metric spaces

We provide new characterizations of Sobolev ad BV spaces in doubling and Poincar\ue9 metric spaces in the spirit of the Bourgain\u2013Brezis\u2013Mironescu and Nguyen limit formulas holding in domains of R^N

Robust exponential attractors for a family of nonconserved phase-field systems with memory

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Soliton dynamics of NLS with singular potentials

We investigate the validity of a soliton dynamics behavior in the semi-classical limit for the nonlinear Schroedinger equation in R^N,N≥3, in presence of a singular external potential

Diffeomorphism-invariant properties for quasi-linear elliptic operators

For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and non-degenerate coerciveness.Comment: 16 page

Concavity properties for quasilinear equations and optimality remarks

In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schr\"odinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $\Omega$},$$ where $\Omega$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $\varphi:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $\varphi(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source.Comment: To be published on Differential and Integral Equation

On fractional Choquard equations

We investigate a class of nonlinear Schrodinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.Comment: revised version, 22 page

Generalized solutions of variational problems and applications

Ultrafunctions are a particular class of generalized functions defined on a hyperreal field that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach