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We study the existence, the stability and the non-degeneracy of normalized standing-waves solutions to a one dimensional non-linear Schr\"odinger equation. The non-linearity belongs to a class of algebraic functions appropriately defined.…

Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain. In this paper, we prove a result of which the following is a by-product: Let $q\in ]0,1[$, $\alpha\in L^{\infty}(\Omega)$, with $\alpha>0$, and $k\in {\bf N}$. Then, the problem…

We study the non-linear minimization problem on $H^1_0(\Omega)\subset L^q$ with $q=\frac{2n}{n-2}$, $\alpha>0$ and $n\geq4$~: \[\inf_{\substack{u\in H^1_0(\Omega) \|u\|_{L^q}=1}}\int_\Omega a(x,u)|\nabla u|^2 - \lambda \int_{\Omega}…

The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schr\'[o]dinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see…

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^2$. For $\epsilon>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $-\Delta u=\frac{1}{\epsilon^2}(1-|u|^2)u$ in $\Omega$ such that on $\partial \Omega$ u…

This paper concerns the Cauchy problem in R^d for the stochastic Navier-Stokes equation \partial_tu=\Delta u-(u,\nabla)u-\nabla p+f(u)+ [(\sigma,\nabla)u-\nabla \tilde p+g(u)]\circ \dot W, u(0)=u_0,\qquad divu=0, driven by white noise \dot…

In this paper, we propose a unified non-conforming least-squares spectral element approach for solving Stokes equations with various non-standard boundary conditions. Existing least-squares formulations mostly deal with Dirichlet boundary…

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla…

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 consider the problem $\inf\big\{ \int\!\!\int_\Omega (1 + |\nabla u(x,y)|^2)^{-1} dx dy : \text{ the function } u : \Omega \to \mathbb{R} \text{ is concave and } 0 \le u(x,y) \le M \text{ for all } (x,y) \in \Omega =\{ (x,y): x^2 + y^2…

We consider the problem \[-\Delta u + W(x)u = ((1/{|x|^{\alpha}} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(\Omega)\], where $\Omega$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $\alpha \in(0,N)$, $p\in[2,(2N-\alpha)/(N-2)$, $W$ is…

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 consider the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq…

In this work, we study the minimization of nonlinear functionals in dimension $d\geq 1$ that depend on a degenerate radial weight $w$. Our goal is to prove the existence of minimizers in a suitable functional class here introduced and to…

There are two cases when the nonlinear Schr\"odinger equation (NLSE) with an external complex potential is well-known to support continuous families of localized stationary modes: the ${\cal PT}$-symmetric potentials and the Wadati…

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^{\alpha}}\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…

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 prove that there exists a weak solution of the Stokes system with a non-zero external force and no-slip boundary conditions in a half space of dimensions three and higher so that its normal derivatives are unbounded near boundary. A…

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla…