Related papers: On translation invariant constrained minimization …

Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \geq 3 $. Given $p_0 \in M$, $\lambda \in \mathcal{R}$ and $\sigma \in (0,2]$, we study existence and non existence of minimizers of the following quotient:…

In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schr\"odinger equation $$(-\Delta)^s u + \mu u = g(u) \quad \hbox{in $\mathbb{R}^N$},$$ under the mass constraint…

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the…

The present work constitutes a first step towards establishing a systematic framework for treating variational problems that depend on a given input function through a mixture of its derivatives of different orders in different directions.…

Consider nonlinear Schr\"odinger equations with small nonlinearities \[\frac{d}{dt}u+i(-\triangle u+V(x)u)=\epsilon \mathcal{P}(\triangle u,u,x),\quad x\in \mathbb{T}^d.\eqno{(*)}\] Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis…

In this paper we study $2$nd order $L^\infty$ variational problems, through seeking to minimise a supremal functional involving the Hessian of admissible functions as well as lower-order terms. Specifically, given a bounded domain…

We study the problem (-\epsilon\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=\epsilon ^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}\epsilon\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where…

We study the minimizer of the electrostatic Born--Infeld energy \begin{equation*} \int_{\mathbb{R}^n}1-\sqrt{1-|D v|^2}\ dx-\int_{\mathbb{R}^n}\rho v\ dx, \end{equation*} which vanishes at infinity. We show that the minimizer $u$ is…

The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $N$ lowest eigenvalues of a Schr\"odinger operator $-\Delta-V(x)$ in terms of an $L^p(\mathbb{R}^d)$ norm of the potential $V$. We prove here the existence…

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a…

We look for solutions to the Schr\"odinger equation \[ -\Delta u + \lambda u = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = \rho^2$, with $N\ge2$. The behaviour of $g$ at the origin…

We consider the problem of minimizing the entropy of a law with respect to the law of a reference branching Brownian motion under density constraints at an initial and final time. We call this problem the branching Schr\"odinger problem by…

This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schr\"odinger problem, Brenier's relaxed model for incompressible fluids, the so-called Br\"odinger…

We prove the local H\"{o}lder continuity of strong local minimizers of the stored energy functional \[E(u)=\int_{\om}\lambda |\nabla u|^{2}+h(\det \nabla u) \,dx\] subject to a condition of `positive twist'. The latter turns out to be…

In this work, we study the two following minimization problems for $r \in \mathbb{N}^{*}$, \begin{equation*} \begin{array}{ccc} S_{0,r}(\varphi)=\displaystyle\inf_{u\in H_{0}^{r}(\Omega)\,|u+\varphi\|_{L^{2^{*r}}}=1}\|u\|_{r}^{2}&…

We investigate the properties of standing waves to a nonlinear Schr\"odinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground…

In this result, we develop the techniques of \cite{KS1} and \cite{BW} in order to determine a class of stable perturbations for a minimal mass soliton solution of a saturated, focusing nonlinear Schr\"odinger equation {c} i u_t + \Delta u +…

The paper studies the existence of minimizers for Rayleigh quotients $\mu_{\Omega}=\inf\frac{\int_\Omega|\nabla u|^2}{\int_\Omega V{|u|^2}} $, where $\Omega$ is a domain in $\mathbb{R}^N$, and $V$ is a nonzero nonnegative function that may…

This paper is motivated by a gauged Schr\"odinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $$ - \Delta u(x) + \left(\omega +…

The inverse problem of determining the unknown potential $f>0$ in the partial differential equation $$\frac{\Delta}{2} u - fu =0 \text{ on } \mathcal O ~~\text{s.t. } u = g \text { on } \partial \mathcal O,$$ where $\mathcal O$ is a bounded…