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Simone Secchi

Publications and source records attributed to Simone Secchi.

50 records · Page 3Linked to original sources
arXiv2011

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^α}\ast |u|^{p}) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}%] where $A\ $is a real valued vector potential, $V$ is a real valued scalar potential$,$ $N\geq3$, $α\in(0,N)$ and $2-(α/N) <p<(2N-α)/(N-2)$. \ We assume that both $A$ and $V$ are compatible with the action of some group $G$ of linear isometries of $\mathbb{R}^{N}$. We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition \[ u(gx)=τ(g)u(x)\text{\ \ \ for all}g\in G,\text{}x\in\mathbb{R}^{N}, \] where $τ:G\rightarrow\mathbb{S}^{1}$ is a given group homomorphism into the unit complex numbers.

math.AP↗
arXiv2009

A note on Schrödinger--Newton systems with decaying electric potential

We prove the existence of solutions for the singularly perturbed Schrödinger--Newton system {ll} \hbar^2 Δψ- V(x) ψ+ U ψ=0 \hbar^2 ΔU + 4πγ|ψ|^2 =0 . \hbox{in $\mathbb{R}^3$} with an electric potential (V) that decays polynomially fast at infinity. The solution $ψ$ concentrates, as $\hbar \to 0$, around (structurally stable) critical points of the electric potential. As a particular case, isolated strict extrema of (V) are allowed.

math.AP↗
arXiv2008

Multi-peak solutions for magnetic NLS equations without non--degeneracy conditions

In the work we consider the magnetic NLS equation (\frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u = 0 \quad {in} \R^N where $N \geq 3$, $A \colon \R^N \to \R^N$ is a magnetic potential, possibly unbounded, $V \colon \R^N \to \R$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\colon \R^N \to \C$, under conditions on the nonlinearity which are nearly optimal.

math.AP↗
arXiv2008

Multiple Solutions for a Henon-Like Equation on the Annulus

For the equation (-Δu = | |x|-2 |^αu^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (α) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A symmetry--breaking phenomenon appears, showing that the least--energy solutions cannot be radial functions.

math.AP↗
arXiv2007

A note on the radial solutions for the supercritical Henon equation

We prove the existence of a positive radial solution for the Hénon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.

math.AP↗
arXiv2006

Morse index properties of colliding solutions to the $N$-body problem

We study a singular Hamiltonian system with an $\al$-homogeneous potential that contains, as a particular case, the classical $N$--body problem. We introduce a variational Morse--like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories under suitable spectral conditions provided $\al$ is sufficiently away from zero. We then prove some minimality results for small values of the parameter $\al$.

math.DS↗
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↗