Related papers: Normalized solutions and stability for biharmonic …

We analyze $L^2$-normalized solutions of nonlinear Schr\"odinger systems of Gross-Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary conditions. We provide sufficient conditions for the existence of orbitally stable…

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0. $$ The equation has a two-parameter family of…

In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schr\"odinger equation $$ \left\{ \begin{array}{ll} -\Delta u = \lambda u + f(u)\quad & \text{in…

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 look for solutions to the following coupled Schr\"{o}dinger system \begin{equation*} \begin{cases} -\Delta u+\lambda_{1}u=\alpha_{1}|u|^{p-2}u+\mu_{1}u^{3}+\rho v^{2}u & \text{in} \ \ \mathbb{R}^{N}, -\Delta…

We consider the standing-wave problem for a nonlinear Schr\"{o}dinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a…

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…

The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schr\"odinger system: $$i\partial_tu +\partial_x^2u-\partial_x^4u=\lambda |u|^2u,$$…

We consider a derivative nonlinear Schr\"odinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the…

On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge3$, we consider the stationary Schr\"odinger equation $\Delta_gu+h_0u=\left|u\right|^{2^*-2}u$, where $\Delta_g:=-\text{div}_g\nabla$, $h_0\in C^1\left(M\right)$…

We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in $\mathbb{R}^{N}$ \begin{equation*} \Delta^{2}u+V(\varepsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{2^{**}-2}u…

We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times H^1(\R^N)$ for systems of nonlinear Schr\"odinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz…

This paper is concerned with the existence of solutions to the problem $$-\left(a+ b\int_{\mathbb{R}^{N}}|\nabla u|^{2} dx \right)\Delta u +V(x)u+\lambda u = |u|^{p-2}u,\ \ x \in \mathbb{R}^{N},\ \ \lambda \in \mathbb{R}^{+} $$ where $a,…

In this paper, we study the following semilinear Schr\"odinger system $$ -\triangle u+u=(1+K_\alpha(\epsilon x))|u|^{p-2}u\ in \mathbb{R}^N, u\in H^1(\mathbb{R}^N) $$ where $3\leq p<2^*$ and $\epsilon>0$, $\alpha>0$ are small parameters.…

We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the…

We consider the following generalized derivative nonlinear Schr\"odinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2\sigma}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $\sigma\in(0,1)$. The…

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,…

We study the instability of standing-wave solutions $e^{i\omega t}\phi_{\omega}(x)$ to the inhomogeneous nonlinear Schr\"{o}dinger equation $$i\phi_t=-\triangle\phi+|x|^2\phi-|x|^b|\phi|^{p-1}\phi, \qquad \in\mathbb{R}^N, $$ where $ b > 0 $…

We are looking for solutions to nonlinear Schr\"odinger-type equations of the form $$ (-\Delta)^{\alpha / 2} u (x) + V(x) u(x) = h (x,u(x)), \quad x \in \mathbb{R}^N, $$ where $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is an external…

We look for ground state solutions to the following nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x)u = f(x,u)-\Gamma(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$$ where $V=V_{per}+V_{loc}\in L^{\infty}(\mathbb{R}^N)$ is the sum of a periodic…