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We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical…

We study the existence of positive solutions on $\R^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f(u)$ where $N\geq 1$ and $f$ is modeled on the power case $f(u)=|u|^{p-1}u$. Denoting with $c$ the mountain pass level of $\f(u)=\tfrac…

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…

We study the following nonlinear critical curl-curl equation \begin{equation}\label{eq0.1}\nabla\times \nabla\times U +V(x)U=|U|^{p-2}U+ |U|^4U,\quad x\in \mathbb{R}^3,\end{equation} where $V(x)=V(r, x_3)$ with $r=\sqrt{x_1^2+x_2^2}$ is…

We study the following nonlinear Schr\"{o}dinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta…

In \cite{duck-merle}, T. Duyckaerts and F. Merle studied the variational structure near the ground state solution $W$ of the energy critical NLS and classified the solutions with the threshold energy $E(W)$ in dimensions $d=3,4,5$ under the…

We prove existence and qualitative properties of ground state solutions to a generalized nonlocal 3rd-4th order Gross-Pitaevskii equation. Using a mountain pass argument on spheres and constructing appropriately localized Palais-Smale…

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $3\leq p<5$. We generalize inward/outward energy theory and weighted…

We consider the Cauchy-problem for a parabolic equation of the following type: \begin{equation*} \frac{\partial u}{\partial t}= \Delta u+ f(u,|x|), \end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial…

The aim of this article is to study a nonlinear system modeling a Non-Newtonian fluid of polymer aqueous solutions. We are interested here in the existence of weak solutions for the stationary problem in a bounded plane domain or in…

We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-\Delta u_j-\frac{\lambda_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}\beta_{jk}\alpha_{jk}u_j^{\alpha_{jk}-1}u_k^{\alpha_{kj}},\;\;x\in\R^N,…

We investigate normalized solutions for a class of nonlinear Schr\"{o}dinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+\beta |u|^{p-2}u$ on a bounded domain $\Omega$. Firstly, when…

In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schr\"odinger equation $iu_t + \Delta u =|x|^{-b} |u|^\alpha u$ in four space dimension, where $s_c…

In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on $\mathbb{R}^d$ ($d\geq 5$) with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in…

We consider the semilinear fractional equation $ (I-\Delta)^s u = a(x) |u|^{p-2}u$ in $\mathbb{R}^N$, where $N \geq 3$, $0<s<1$, $2<p<2N/(N-2s)$ and $a$ is a bounded weight function. Without assuming that $a$ has an asymptotic profile at…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$ where $L_\mu=\Delta +…

In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$,…

We investigate the existence of solutions to the fractional nonlinear Schr\"{o}dinger equation $(-\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly…

Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schr\"odinger equation $iu_t+\Delta u=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$.…

We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover…