Related papers: Ground states for the Hartree energy functional in…

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…

We prove the existence of the set of ground states in a suitable energy space $\Sigma^s=\{u: \int_{\mathbb{R}^N} \bar{u}(-\Delta+m^2)^s u+V |u|^2<\infty\}$, $s\in (0,\frac{N}{2})$ for the mass-subcritical nonlinear fractional Hartree…

Consider the Hartree-type equation in $\mathbb{R}^3$ with a delta potential formally described by $$ i \partial_t \psi = - \Delta_x \psi + \alpha \delta_0 \psi - (I_\beta \ast |\psi|^p) |\psi|^{p - 2} \psi $$ where $\alpha \in \mathbb{R}$;…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[\sqrt{-\Delta+m^2}\, u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $V$ is a bounded…

This paper is concerned with the ground states of a coupled pseudo-relativistic Hartree system in $\mathbb{R} ^{3} $ with trapping potentials, where the intraspecies and the interspecies interaction are both attractive. By investigating an…

We prove the existence of a ground state for some variational problems in Hilbert spaces, following the approach of Berestycki and Lions. Next, we examine the problem of constructing ground state solutions…

We study a class of mixed local-nonlocal equations with Hartree-type nonlinearities of the form \begin{equation}\label{meqnab} -\Delta u + (-\Delta)^s u + u = (I_\alpha * F(u))\,F'(u) \quad \text{in } \mathbb{R}^N, \end{equation} where $N…

The dynamic properties of normalized ground states for the Hartree equation with a harmonic potential are addressed. The existence of normalized ground state for any prescribed mass is confirmed according to mass-energy constrained…

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-\Delta+m^2)^\sigma u+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<\sigma<1$, $V$ is a…

In this paper, we study the following coupled nonlinear logarithmic Hartree system \begin{align*} \left\{ \displaystyle \begin{array}{ll} \displaystyle -\Delta u+ \lambda_1 u =\mu_1\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2 \right)u+\beta…

We study nondegeneracy of ground states of the Hartree equation $$ -\Delta u+u=(I_{2}\ast u^2)u\quad\mbox{ in }\mathbb R^n $$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a…

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that…

In any dimension $N \geq 1$, for given mass $m > 0$ and for the $C^1$ energy functional \begin{equation*} I(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\int_{\mathbb{R}^N}F(u)dx, \end{equation*} we revisit the classical problem of…

We consider minimizers of the following mass critical Hartree minimization problem: \[ e_\lambda(N):=\underset{\{u\in H^1(R^d),\,\|u\|^2_2=N\}}{\inf} E_\lambda(u),\,\ d\ge 3, \] where the Hartree energy functional $E_\lambda(u)$ is defined…

We investigate the ground states for the focusing, subcritical nonlinear Schr\"odinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for…

We consider ground states of $L^2$-subcritical nonlinear Schr\"{o}dinger equation (1.1), which can be described equivalently by minimizers of the following constraint minimization problem $$ e(\rho):=\inf\{E_{\rho}(u):u\in…

We consider the generalized Choquard equation describing trapped electron gas in 3 dimensional case. The study of orbital stability of the energy minimizers (known as ground states) depends essentially in the local uniqueness of these…

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad…

Using a straightforward extension of the analysis of Lieb and Wu, we derive a simple analytic form for the ground state energy of a one-dimensional Hubbard ring in the atomic limit. This result is valid for an \textit{arbitrary} number of…

In any dimension $N \geq 1$, for given mass $a>0$, we look to critical points of the energy functional $$ I(u) = \frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2 dx + \int_{\mathbb{R}^N}u^2|\nabla u|^2 dx - \frac{1}{p}\int_{\mathbb{R}^N}|u|^p…