Eugenio Montefusco

Semiclassical states for weakly coupled nonlinear Schrödinger systems

We consider systems of weakly coupled Schrodinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

On the blow-up threshold for weakly coupled nonlinear Schrödinger equations

We study the Cauchy problem for a system of two coupled nonlinear focusing Schrodinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we prove, for suitable values of the parameters, that the blow-up threshold (if the nonlinearity has the critical growth) is a universal constant.

Nonlinear eigenvalue problems for quasilinear operators on unbounded domains

Abstract. We prove several existence results for eigenvalue problems involving the p-Laplacian and a nonlinear boundary condition on unbounded domains. We treat the non-degenerate subcritical case and the solutions are found in an appropriate weighted Sobolev space.

Orbital Stability Property for Coupled Nonlinear Schrodinger Equations

Abstract Orbital stability property for weakly coupled nonlinear Schrödinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable Schrödinger weakly coupled system, even if they are not ground states.

ON THE LOGARITHMIC SCHRÖDINGER EQUATION

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system

Existence of radial solutions with a prescribed number of nodes is established, via variational methods, for a system of weakly coupled nonlinear Schrodinger equations. The main goal is to obtain a nodal solution with all vector components not identically zero and an estimate on their energies.

On the shape of blow-up solutions to a mean field equation

We analyse the structure of non-radial N-point blow up solutions sequences for the Liouville type equation on the two-dimensional unit disc, In the case N = 1, 2, we provide necessary and sufficient conditions for the existence of blow up solutions and, in the spirit of Chen and Lin (2001 Ann. Inst. H. Poincare. Anal. Non Lineare 18 271), prove their axial symmetry with respect to the diameter joining the maximum points. Finally, we prove that a non-radial one point blow up solution exists only if λ − 8π > 0.

Soliton dynamics for CNLS systems with potentials

The semiclassical limit of a weakly coupled nonlinear focusing Schrodinger system in presence of a nonconstant potential is studied. The initial data is of the form (u1,u2) with ui = ri( x−x e )e (i/e)x·ξ , where (r1, r2) is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For e sufficiently small, the solution (φ1,φ2) will been shown to have, locally in time, the form (r1( x−x(t) e )e (i/e)x·ξ(t), r2( x−x(t) e )e (i/e)x·ξ(t)), where (x(t), ξ(t)) is the solution of the Hamiltonian system ẋ(t) = ξ(t), ξ(t) = −∇V (x(t)) with x(0) = x and ξ(0) = ξ.

Oscillating solutions for nonlinear Helmholtz equations

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behaviour at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein–Gordon or Schrödinger equations with large frequencies.

Semilinear Hamiltonian Schrödinger systems

In this paper we investigate on local and global existence for some semilinear Schrodinger systems having conservation of the energy and masses. Moreover we presents some blowing up examples for 2 × 2 systems.