Related papers: Normalized solutions and mass concentration for su…

We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(\epsilon x)u=\lambda u+\left(I_\alpha *|u|^q\right)|u|^{q-2} u+\left(I_\alpha *|u|^p\right)|u|^{p-2} u, \quad…

In the present paper, we prove the existence of solutions $(\lambda_1,\lambda_2,u,v)\in\mathbb{R}^2\times H^1(\mathbb{R}^3,\mathbb{R}^2)$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1u=\mu_1 u^3+\beta…

In this paper, we show that any solution of the nonlinear Schr{{\"o}}dinger equation $iu\_t+\Delta u\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is…

We consider the following nonlinear Schr\"odinger equations with critical growth: \begin{equation} - \Delta u + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial…

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schr\"odinger equation $$ -\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(\Omega),\quad\int_\Omega u^2dx=\rho^2,\quad\lambda\in\mathbb{R}, $$…

We consider the following two-component coupled nonlinear Schr\"odinger (CNLS) system: \[ \begin{cases} -\Delta u +(P(x) + \lambda ) u=\mu_1 u^3+\beta u v^2, & \text{in } \mathbb{R}^N,\\ -\Delta v +(Q(x) + \lambda ) v =\mu_2 v^3+\beta vu^2,…

We consider the problem $-\Delta u+\lambda u=u^{p-1}$, where $u\in H^1_0(\Omega)$ verifies $\|u\|_{L^2}=m>0$, and $\lambda\in [0,+\infty)$. Here, $\mathbb{R}^N\setminus\Omega$ is nonempty and compact. We prove the existence of a solution…

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schr\"odinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$ -\text{div}(|x|^{-2a}\nabla u)=\lambda…

In this paper, we study the following nonlinear Schr\"{o}dinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -\Delta u+V(x)u=\partial_v H(x,u,v)+\omega v, \ x \in \mathbb{R}^N, \\ -\Delta v+V(x)v=\partial_u…

In this paper, we consider the existence and multiplicity of normalized solutions for the following $p$-Laplacian critical equation \begin{align*} \left\{\begin{array}{ll} -\Delta_{p}u=\lambda\lvert u\rvert^{p-2}u+\mu\lvert…

In this paper, we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger systems with exponential growth \begin{align*} \left\{ \begin{aligned} &-\Delta u+\lambda_{1}u=H_{u}(u,v), \quad \quad \hbox{in…

In this paper, we study the normalized solutions of the Schr\"{o}dinger system with trapping potentials \begin{equation}\label{eq:diricichlet} \begin{cases} -\Delta u_1+V_1(x)u_1-\lambda_1 u_1=\mu_1 u_1^3+\beta u_1u_2^{2}+\kappa…

We investigate normalized solutions for a class of nonlinear Schr\"{o}dinger (NLS) equations with potential $V$ and inhomogeneous nonlinearity $g(|u|)u=|u|^{q-2}u+\beta |u|^{p-2}u$ on a bounded domain $\Omega$. Firstly, when…

In this paper, we are concerned with the coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} -\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta v^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta…

We consider finite time blowup solutions of the $L^2$-critical cubic focusing nonlinear Schr\"odinger equation on $\R^2$. Such functions, when in $H^1$, are known to concentrate a fixed $L^2$-mass (the mass of the ground state) at the point…

In this paper, we prove the existence of normalized solutions for the following Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u-\lambda u=f(u), & x\in \R^N, \int_{\R^N}u^2\mathrm{d}x=c \end{array} \right.…

In this paper, we prove the existence of positive solutions $(\lambda_1,\lambda_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schr\"odinger system $$\begin{cases} -\Delta u + \lambda_1 u= \mu_1|u|^{p-2}u+\beta v \quad…

Goal of this paper is to study positive semiclassical solutions of the nonlinear Schr\"odinger equation $$ \varepsilon^{2s}(- \Delta)^s u+ V(x) u= f(u), \quad x \in \mathbb{R}^N,$$ where $s \in (0,1)$, $N \geq 2$, $V \in…

We get multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation $$ (-\Delta)^su+V(\varepsilon x)u=\lambda u+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where…

We are concerned with the normalized $\ell$-peak solutions to the nonlinear Schr\"{o}dinger equation \[ -\varepsilon^2\Delta v+V(x)v=f(v)+\lambda v,\quad \int_{\mathbb{R}^N}v^2 =\alpha \varepsilon^N. \] Here $\lambda \in \mathbb{R}$ will…