Related papers: Schr\"odinger-Poisson systems with a general criti…
We consider a fractional Schr\"odinger-Poisson system in the whole space $\mathbb R^{N}$ in presence of a positive potential and depending on a small positive parameter $\varepsilon.$ We show that, for suitably small $\varepsilon$ (i.e. in…
We look for positive solutions to the nonlinear Schrodinger equation with a potential, under the hypothesis of zero mass on the nonlinearity, in a particular situation. Existence and multiplicity results are provided.
We demonstrate existence of positive bound and ground states for a system of coupled nonlinear Schr\"odinger--Korteweg-de Vries equations. More precisely, we prove there is a positive radially symmetric ground state if either the coupling…
In this paper we consider the problem \begin{equation*} \left \{ \begin{array}{l} -\Delta u \pm \phi u + W'(x,u) = 0\hbox{ in } \mathbb{R}^2,\newline \Delta \phi = u^2 \hbox{ in } \mathbb{R}^2, \end{array} \right. \end{equation*} where $W$…
In this paper we establish, using variational methods combined with the Moser-Trudinger inequality, existence and multiplicity of weak solutions for a class of critical fractional elliptic equations with exponential growth without a…
We study the existence of positive solutions for nonlocal systems in gradient form and set in the whole $\mathbb R^N$. A quasilinear fractional Schr\"odinger equation, where the leading operator is the $\frac Ns$-fractional Laplacian, is…
In dimension two, we investigate a free energy and the ground state energy of the Schr\"odinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling…
In this paper we carry on the study of a system recently introduced by the first author as the planar version of the well known electrostatic Schr\"odinger - Maxwell equations. In the positive potential case, we exhibit situations where the…
In this paper, we consider the multi-species nonlinear Schr\"odinger systems in $\bbr^N$: \begin{equation*} \left\{\aligned&-\Delta u_j+V_j(x)u_j=\mu_ju_j^3+\sum_{i=1;i\not=j}^d\beta_{i,j} u_i^2u_j\quad\text{in }\bbr^N,…
We prove the global existence of solution to the small data mass critical stochastic nonlinear Schr\"{o}dinger equation in $d=1$. We further show the stability of the solution under perturbation of initial data. Our construction starts with…
We consider the existence of solutions for nonlinear Schr\"odinger equations on noncompact metric graphs with localized nonlinearities. In an $L^2$-supercritical regime, we establish the existence of infinitely many solutions for any…
We study the stationary states of the semi-relativistic Schr\"odinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states…
We consider the Schr\"odinger-Poisson system in the attractive (plasma physics) Coulomb case. Given a steady state from a certain class we prove its nonlinear stability, using an appropriately defined energy-Casimir functional as Lyapunov…
We study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time-independent Schr\"odinger equations $$ - \Delta u_j+ \lambda_j u_j =\mu_j u_j^3+ \sum_{k=1;k\neq j}^N\beta_{jk} u_ju_k^2,\quad…
We consider the Cauchy problem for quadratic nonlinear Klein-Gordon systems in two space dimensions with masses satisfying the resonance relation. Under the null condition in the sense of J.-M. Delort, D. Fang, R. Xue (2004), we show the…
This paper is concerned with the global existence of small solutions to pure-power nonlinear Schroedinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where…
In this paper, we consider the existence of positive solutions with prescribed $L^2$-norm for the following nonlinear Schr\"{o}dinger equation involving potential and Sobolev critical exponent \begin{equation*} \begin{cases} -\Delta…
The initial value problem is considered for a higher order nonlinear Schr\"odinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional…
We present a numerical study of a derivative nonlinear Schr\"odinger equation with a general power nonlinearity, $|\psi|^{2\sigma}\psi_x$. In the $L^2$-supercritical regime, $\sigma>1$, our simulations indicate that there is a finite time…
In this paper, we consider the following nonlinear critical Schr\"odinger system: \begin{eqnarray*}\begin{cases} -\Delta u=K_1(y)u^{2^*-1}+\frac{1}{2} u^{\frac{2^*}{2}-1}v^\frac{2^*}{2}, \,\,\,\,\,y\in\Omega,\,\,\,\,\,u>0,\cr -\Delta…