Related papers: Standing waves for defocusing nonlinear Schr\"odin…
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}+…
We study analytically and numerically the stability of the standing waves for a nonlinear Schr\"odinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the…
We study the existence and stability of standing waves for a system of nonlinear Schr\"odinger equations with quadratic interaction in dimensions $d\leq 3$. We also study the characterization of finite time blow-up solutions with minimal…
We are concerned with the following Schr\"odinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u + \psi u = \lambda |u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill -…
We study the mixed dispersion fourth order nonlinear Schr\"odinger equation \begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma \Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \R \times\R^N,…
In this paper, we investigate the existence of positive solution for the following class of elliptic equation $$ - \epsilon^{2}\Delta u +V(x)u= f(u) \,\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon >0$ is a positive parameter,…
We are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…
We study the stability/instability of standing waves for the one dimensional nonlinear Schr\"odinger equation with double power nonlinearities: \begin{align*} &i\partial_t u +\partial_x^2 u -|u|^{p-1}u +|u|^{q-1}u=0, \quad (t,x)\in…
We investigate the asymptotic stability of standing waves for a model of Schr\"odinger equation with spatially concentrated nonlinearity in space dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point…
We study standing waves for the nonlinear Schr\"odinger equation on a discrete graph. We characterize for a self-adjoint realizations of Schr\"odinger operators conditions related with the geometry of the graph that guarantee discreteness…
We study the quasi-periodic standing wave solutions of the focusing and defocusing cubic nonlinear Schr{\"o}dinger equations in dimension one. In the defocusing case, we establish a diffeomorphic correspondence between the invariants of the…
We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schr\"odinger--Newton system with a point interaction: \[ \begin{cases} - \Delta_\alpha u = w u + \beta u |u|^{p - 2} &\text{on} ~…
The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr\"{o}dinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the…
In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schr\"odinger equations with potentials. We show that the standing waves are orbitally stable for all frequencies in the $L^{2}$- subcritical and…
In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schr\"{o}dinger equation \begin{equation} i\partial_{t}\psi=(-\Delta)^{s}\psi-f(\psi), \qquad (0.1)\end{equation} where…
On a star graph made of $N \geq 3$ halflines (edges) we consider a Schr\"odinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there…
In this paper, by combing the variational methods and Trudinger-Moser inequality, we study the existence and multiplicity of the positive standing wave for the following Chern-Simons-Schr\"odinger equation \begin{equation} -\Delta u+u…
This paper is motivated by a gauged Schr\"{o}dinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational…
In this article, we study the standing-wave solutions to a class of systems of nonlinear Schr\"odinger equations. Our target is all the standard forms of the NLS systems, with two unknowns, that have a common linear part and cubic…
In this work, we study the existence and orbital (in)stability of certain standing-wave solutions for the cubic nonlinear Schr\"odinger equation (NLS) posed on a looping-edge graph $\mathcal{G}$, consisting of a circle and a finite number…