172 research outputs found

### Ground states for the pseudo-relativistic Hartree equation with external potential

We prove existence of positive ground state solutions to the
pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{
\begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta}
\right)|u|^{\theta -2} u \quad\text{in $\mathbb{R}^N$}\\ u \in
H^{1/2}(\mathbb{R}^N) \end{array} \right. \end{equation*} where $N \geq 3$, $m
>0$, $V$ is a bounded external scalar potential and $W$ is a convolution
potential, radially symmetric, satisfying suitable assumptions. We also furnish
some asymptotic decay estimates of the found solutions.Comment: In pres

### Semiclassical analysis for pseudo-relativistic Hartree equations

In this paper we study the semiclassical limit for the pseudo-relativistic
Hartree equation $\sqrt{-\varepsilon^2 \Delta + m^2}u + V u = (I_\alpha *
|u|^{p})
|u|^{p-2}u$ in $\mathbb{R}^N$ where $m>0$, $2 \leq p < \frac{2N}{N-1}$, $V
\colon \mathbb{R}^N \to \mathbb{R}$ is an external scalar potential, $I_\alpha
(x) = \frac{c_{N,\alpha}}{|x|^{N-\alpha}}$ is a convolution kernel,
$c_{N,\alpha}$ is a positive constant and $(N-1)p-N<\alpha <N$. For $N=3$,
$\alpha=p=2$, our equation becomes the pseudo-relativistic Hartree equation
with Coulomb kernel.Comment: Accepted for publication by Journal of Differential Equation

### 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.Comment: Important modification in the last part of the pape

### Nonlinear Schr{\"o}dinger equation: concentration on circles driven by an external magnetic field

In this paper, we study the semiclassical limit for the stationary magnetic
nonlinear Schr\"odinger equation \begin{align}\label{eq:initialabstract}\left(
i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in
\mathbb{R}^{3},\end{align}where p\textgreater{}2, $A$ is a vector potential
associated to a given magnetic field $B$, i.e $\nabla \times A =B$ and $V$ is a
nonnegative, scalar (electric) potential which can be singular at the origin
and vanish at infinity or outside a compact set.We assume that $A$ and $V$
satisfy a cylindrical symmetry. By a refined penalization argument, we prove
the existence of semiclassical cylindrically symmetric solutions of upper
equation whose moduli concentrate, as $\hbar \to 0$, around a circle. We
emphasize that the concentration is driven by the magnetic and the electric
potentials. Our result thus shows that in the semiclassical limit, the magnetic
field also influences the location of the solutions of
(\ref{eq:initialabstract}) if their concentration occurs around a locus, not
a single point

### On the Poincaré-Hopf Theorem for Functionals Defined on Banach Spaces

Abstract
Let X be a reflexive Banach space and f : X → ℝ a Gâteaux differentiable function with f′ demicontinuous and locally of class (S)+. We prove that each isolated critical point of f has critical groups of finite type and that the Poincaré-Hopf formula holds. We also show that quasilinear elliptic equations at critical growth are covered by this result

### 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

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