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Marco Squassina

Publications and source records attributed to Marco Squassina.

144 records · Page 8Linked to original sources
arXiv2009

Soliton dynamics for the nonlinear Schrödinger equation with magnetic field

The semiclassical limit of a nonlinear focusing Schrödinger equation in presence of nonconstant electric and magnetic potentials V,A is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation. The concentration curve of the solutions is a parameterization of the solutions of a Newton ODE involving the electric force as well as the magnetic force via the Lorenz law of electrodynamics.

math.AP↗
arXiv2009

Symmetry results for nonvariational quasi-linear elliptic systems

By virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.

math.AP↗
arXiv2008

Asymptotic behavior of a thermoviscoelastic plate with memory effects

We consider a coupled linear system describing a thermoviscoelastic plate with hereditary effects. The system consists of a hyperbolic integrodifferential equation, governing the temperature, which is linearly coupled with the partial differential equation ruling the evolution of the vertical deflection, presenting a convolution term accounting for memory effects. It is also assumed that the thermal power contains a memory term characterized by a relaxation kernel. We prove that the system is exponentially stable and we obtain a closeness estimate between the system with memory effects and the corresponding memory-free limiting system, as the kernels fade in a suitable sense.

math.AP↗
arXiv2008

Semiclassical states for weakly coupled nonlinear Schrödinger systems

We consider systems of weakly coupled Schrödinger 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.

math.AP↗
arXiv2008

On phase segregation in nonlocal two-particle Hartree systems

We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime.

math.AP↗
arXiv2008

Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems

We consider a system of Gross-Pitaevskii equations in R^2 modelling a mixture of two Bose-Einstein condensates with repulsive interaction. We aim to study the qualitative behaviour of ground and excited state solutions. We allow two different harmonic and off-centered trapping potentials and study the spatial patterns of the solutions within the Thomas-Fermi approximation as well as phase segregation phenomena within the large-interaction regime.

math.AP↗
arXiv2008

On the long term spatial segregation for a competition-diffusion system

We investigate the long term behavior for a class of competition-diffusion systems of Lotka-Volterra type for two competing species in the case of low regularity assumptions on the data. Due to the coupling that we consider the system cannot be reduced to a single equation yielding uniform estimates with respect to the inter-specific competition rate parameter. Moreover, in the particular but meaningful case of initial data with disjoint support and Dirichlet boundary data which are time-independent, we prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially segregated state satisfying some variational inequalities.

math.AP↗
arXiv2005

Locating the peaks of semilinear elliptic systems

We consider a system of weakly coupled singularly perturbed semilinear elliptic equations. First, we obtain a Lipschitz regularity result for the associated ground energy function $Σ$ as well as representation formulas for the left and the right derivatives. Then, we show that the concentration points of the solutions locate close to the critical points of $Σ$ in the sense of subdifferential calculus.

math.AP↗
arXiv2004

On the location of spikes for the Schrodinger equation with electromagnetic field

We consider the standing wave solutions of the three dimensional semilinear Schrodinger equation with competing potential functions $V$ and $K$ and under the action of an external electromagnetic vector field $A$. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. In the particular but important case of nonlinearities of power type, we prove that the spikes locate at the critical points of a smooth ground energy map independent of $A$.

math.AP↗
arXiv2003

On the location of concentration points for singularly perturbed elliptic equations

By means of a variational identity of Pohožaev-Pucci-Serrin type for solutions of class $C^1$ recently obtained, we give some necessary conditions for locating the concentration points for a class of quasi-linear elliptic problems in divergence form. More precisely we show that the points where the concentration occurs must be critical, either in a generalized or in the classical sense, for a suitable ground state function.

math.AP↗
arXiv2003

Multi-peak solutions for a class of degenerate elliptic equations

By means of a penalization argument due to del Pino and Felmer, we prove the existence of multi-spike solutions for a class of quasilinear elliptic equations under natural growth conditions. Compared with the semilinear case some difficulties arise, mainly concerning the properties of the limit equation. The study of concentration of the solutions requires a somewhat involved analysis in which a Pucci-Serrin type identity plays an important role.

math.AP↗
arXiv2003

Spike solutions for a class of singularly perturbed quasilinear elliptic equations

By means of a penalization scheme due to del Pino and Felmer, we prove the existence of single-peaked solutions for a class of singularly perturbed quasilinear elliptic equations associated with functionals which lack of smoothness. We don't require neither uniqueness assumptions on the limiting autonomous equation nor monotonicity conditions on the nonlinearity. Compared with the semilinear case some difficulties arise and the study of concentration of the solutions needs a somewhat involved analysis in which the Pucci-Serrin variational identity plays an important role.

math.AP↗