相关论文: Standing waves with prescribed mass for the Schr\"…
In the present paper, we study the following Schr\"{o}dinger-Maxwell equation with combined nonlinearities \begin{align*} \displaystyle - \Delta u+\lambda u+ \left(|x|^{-1}\ast |u|^2\right)u =|u|^{p-2}u +\mu|u|^{q-2}u\quad \text{in} \…
We investigate standing waves with prescribed mass for a class of Schrodinger equations with competing Van Der Waals type potentials, arising in a model of non-relativistic bosonic atoms and molecules. By developing an approach based on a…
The paper deals with the existence of standing wave solutions for the Schr\"odinger-Poisson system with prescribed mass in dimension $N=2$. This leads to investigate the existence of normalized solutions for an integro-differential equation…
In this paper, a nonlinear Schr\"odinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both…
Consider the Neumann problem: \begin{eqnarray*} \begin{cases} &-\Delta u-\frac{\mu}{|x|^2}u +\lambda u =|u|^{q-2}u+|u|^{p-2}u ~~~\mbox{in}~~\mathbb{R}_+^N,~N\ge3, &\frac{\partial u}{\partial \nu}=0 ~~ \mbox{on}~~ \partial\mathbb{R}_+^N…
For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce…
For the cubic Schr\"odinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We…
We study strong instability (instability by blowup) of standing wave solutions for a nonlinear Schr\"odinger equation with an attractive delta potential and $L^2$-supercritical power nonlinearity in one space dimension. We also compare our…
We consider standing waves of the nonlinear Schr\"odinger equation $i\partial_t u = -\Delta_\alpha u + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-\Delta_\alpha$ denotes the Laplacian with a point interaction.…
We consider the nonlinear Schr{\"o}dinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a…
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 study the following singularly perturbed problem for a coupled nonlinear Schr\"{o}dinger system: {displaymath} {cases}-\e^2\Delta u +a(x) u = \mu_1 u^3+\beta uv^2, \quad x\in \R^3, -\e^2\Delta v +b(x) v =\mu_2 v^3+\beta vu^2, \quad x\in…
In this paper, we establish the existence and instability of standing wave for a system of nonlinear Schr\"{o}dinger equations arising in the two-wave model with quadratic interaction in higher space dimensions under mass resonance…
We show the strong instability of radial ground state standing waves for the focusing $L^2$-supercritical nonlinear Schr\"odinger equation with inverse-square potential \[ i\partial_t u + \Delta u + c|x|^{-2} u = - |u|^{\alpha} u, \quad…
In this paper, we consider the following two-component elliptic system with critical growth \begin{equation*} \begin{cases} -\Delta u+(V_1(x)+\lambda)u=\mu_1u^{3}+\beta uv^{2}, \ \ x\in \mathbb{R}^4, -\Delta…
In this paper we investigate the existence of solutions in $H^1(R^N) \times H^1(R^N)$ for nonlinear Schr\"odinger systems of the form \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + r_1\beta…
We consider the focusing nonlinear Schr\"odinger equation with inverse square potential \[ i\partial_t u + \Delta u + c|x|^{-2} u = - |u|^\alpha u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $d \geq…
In this paper, we are concerned with solutions to the following nonlinear Schr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N,…
We investigate standing waves with prescribed mass for a class of biharmonic Schrodinger equations with positive Laplacian dispersion in the Sobolev critical regime. By establishing novel energy inequalities and developing a direct…
We study the following nonlinear Schr\"{o}dinger equation $$ iu_t=-\Delta u+V(x)u-a|u|^qu \quad (t,x)\in \mathbb{R}^1\times \mathbb{R}^2, $$ where $a>0, \ q\in(0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^*:=…