Related papers: On translation invariant constrained minimization …
We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev…
We prove the validity of a small noise large deviation principle for the family of invariant measures $\{\mu_\epsilon\}_{\epsilon>0} $ associated to the one dimensional stochastic Allen-Cahn equation with inhomogeneous Dirichlet boundary…
The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^\alpha |x-y|^{n+2s} |y|^\alpha} \mathrm{d} x \, \mathrm{d} y \geq \Lambda_{n, s, p, \alpha,\beta} \|u…
We establish conditions to ensure the existence of minimizer for a class of mass-constrained functionals involving a nonattractive point interaction in three dimensions. The existence of minimizers follows from the compactness of minimizing…
We show almost sure wellposedness of mild solution to the cubic nonlinear wave equation in a weighted Besov space over $\mathbb R^2$. To achieve this, we show that any weak limit of $\phi^4$ measures on increasing tori is invariant under…
In this work, we study the quasilinear Schr\"{o}dinger equation \begin{equation*} \aligned -\Delta u-\Delta(u^2)u=|u|^{p-2}u+|u|^{q-2}u+\lambda u,\,\, x\in\R^N, \endaligned \end{equation*} under the mass constraint \begin{equation*}…
We study the continuity and the measurability of the solution to Schr\"odinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This…
We study the existence of standing waves for the following weakly coupled system of two Schr\"odinger equations in $\mathbb{R}^N$, $N=2,3$, \[ \begin{cases} i \hslash \partial_{t}\psi_{1}=-\frac{\hslash^2}{2m_{1}}\Delta \psi_{1}+…
This paper investigates the existence of normalized solutions for the following Chern-Simons-Schr\"odinger equation: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left(\frac{h^{2}(\vert x\vert)}{\vert…
Following the original approach introduced by T. Cazenave and P.L. Lions in \cite{CaLi} we prove the existence and the orbital stability of standing waves for the following class of NLS: \label{intr1} i\partial_t u+ \Delta u - V(x) u + Q(x)…
This article is concerned with the quasilinear Schr\"odinger equation \[ \Delta u-\omega u+|u|^{p-1}u+\delta\Delta(|u|^2)u=0, \] where $\delta>0$, $N=2$ and $p>1$ or $N\ge3$ and $1<p<\frac{3N+2}{N-2}$. After proving uniqueness and…
We will study an open problem pertaining to the uniqueness of minimizers for a class of variational problems emanating from Meyer's model for the decomposition of an image into a geometric part and a texture part. Mainly, we are interested…
We study sharp weighted Sobolev-type inequalities of the form \[ \int_{0}^{1}|u(x)|\rho(x) \diff x \leqslant \Lambda \Bigl(\int_{0}^{1}|u^{(k)}(x)|^2 \diff x\Bigr)^{1/2}, \qquad u\in H_0^k(0,1), \] where $\rho$ is a non-negative weight. We…
In this paper, we study the following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=g(u) & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2,\,\,…
We study the regularity of minimizers of the functional $\mathcal E(u):= [u]_{H^s(\Omega)}^2 +\int_\Omega fu$. This corresponds to understanding solutions for the regional fractional Laplacian in $\Omega\subset\mathbb R^N$. More precisely,…
In this article, we study the multiplicity, asymptotics and stability of standing waves with prescribed mass $c>0$ for nonlinear Schr\"odinger equation with rotation in the mass-supercritical regime arising in Bose-Einstein condensation.…
For real bounded functions \Phi and \Psi of compact support, we prove the norm resolvent convergence, as \epsilon and \nu tend to 0, of a family of one-dimensional Schroedinger operators on the line of the form S_{\epsilon, \nu}=…
We find a sign-changing solution for a class of Schr\"odinger-Poisson system in $\mathbb{R}^3$ as an existence result by minimization in a closed subset containing all the sign-changing solutions of the equation. The proof is based on…
This article is concerned with the mathematical analysis of a class of a nonlinear fractional Schrodinger equations with a general Hartree-type integrand. We prove existence and uniqueness of global-in-time solutions to the associated…
We study the least-energy way to reshape a probability distribution when motion is constrained to a horizontal bundle, that is, optimal transport and distribution steering in sub-Riemannian geometry, motivated by density control over…