Related papers: The nonlinear Schr\"odinger equation with combined…

In this paper, we study the scattering for the nonlinear beam equation $u_{tt}+\Delta^2u+mu+\mu |u|^{p-1}u=0$. Our results include two aspects. In the defocusing case we show that the scattering holds for $d=1$, which extends the result in…

We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the…

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schr\"odinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=|u|^{p-2}u \quad\mathrm{in}\ \mathbb{R}^{N},…

In this paper, we consider the defocusing nonlinear Schr\"odinger equation in space dimensions $d\geq 4$. We prove that if $u$ is a radial solution which is \emph{priori} bounded in the critical Sobolev space, that is, $u\in L_t^\infty…

We study the time-asymptotic behavior of solutions of the Schr\"odinger equation with nonlinear dissipation \begin{equation*} \partial _t u = i \Delta u + \lambda |u|^\alpha u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in…

In the present paper, we consider the Cauchy problem of nonlinear Schr\"odinger equations with a derivative nonlinearity which depends only on $\bar{u}$. The well-posedness of the equation at the scaling subcritical regularity was proved by…

We prove global well-posedness and scattering for solutions to the mass-critical inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\pm |x|^{-b}|u|^{\frac{4-2b}{d}}u$ for large $L^2(\mathbb{R} ^d)$ initial data with…

We consider the existence of multiple positive solutions to the nonlinear Schr\"odinger systems sets on $H^1(\mathbb{R}^N) \times H^1(\mathbb{R}^N)$, \[ \left\{ \begin{aligned} -\Delta u_1 &= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…

In present paper, we study the following nonlinear Schr\"{o}dinger equation with combined power nonlinearities \begin{align*} - \Delta u+V(x)u+\lambda u=|u|^{2^*-2}u+\mu |u|^{q-2}u \quad \quad \text{in} \ \mathbb{ R}^N, \ N\geq 3…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in…

We consider the nonlinear stationary Schr\"odinger equation \begin{equation*} -\Delta u -\lambda u= Q(x)|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N \end{equation*} in the case where $N \geq 3$, $p$ is a superlinear, subcritical exponent, $Q$…

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the…

We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing…

We consider the defocusing energy-critical Hartree equation $i\pa_tu+\Delta u=(|\cdot|^{-4}\ast|u|^2)u$ in spatial dimension $d=5$ and prove almost sure scattering with initial data $u_0\in H^s_x(\R^5)$ for any $s\in\R$. The proof relies on…

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $\alpha_\star\leq…

In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: \[ i\frac{d}{dt} \psi^{\varepsilon}(t) =-\Delta\psi^{\varepsilon}(t) +…

We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schr\"odinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-\Delta)^{\frac\alpha 2} u =\pm |u|^\frac{2\alpha}{d}u$, $u(0,\cdot)\in…

We consider the Schr\"odinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +\Delta u=\lambda|u|^{\alpha}u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C} $ with $\Im\lambda<0$. Assuming…

In this paper, we consider the fourth-order Schr\"odinger equations with focusing, $L^2$-supercritical nonlinearity in one dimension. We prove the global existence and scattering of solutions below the ground state threshold under the…

In this article we study some aspects of dispersive and concentration phenomena for the Schr\"odinger equation posed on hyperbolic space $\mathbb{H}^n$, in order to see if the negative curvature of the manifold gets the dynamics more stable…