Related papers: Normalized solutions to mass supercritical Schr\"o…

This paper is devoted to study a class of nonlinear fractional Schr\"{o}dinger equations: \begin{equation*} (-\Delta)^{s}u+V(x)u=f(x,u), \quad \text{in}\: \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $\ N>2s$, $(-\Delta)^{s}$ stands…

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schr\"{o}dinger equation given by \begin{align*} \left\{ \begin{aligned} &-\epsilon^2 \Delta u+V( x)u=\lambda u+u \log u^2,…

In this paper we study the existence of positive normalized solutions of the following coupled Schr\"{o}dinger system: \begin{align} \left\{ \begin{aligned} & -\Delta u = \lambda_u u + \mu_1 u^3 + \beta uv^2, \quad x \in \Omega, \\ &…

We study the existence of normalized solutions to the following logarithmic Schr\"{o}dinger equation \begin{equation*}\label{eqs01} -\Delta u+\lambda u=\alpha u\log u^2+\mu|u|^{p-2}u, \ \ x\in\R^N, \end{equation*} under the mass constraint…

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-\Delta)^s u+V(\varepsilon x)u=\lambda…

In the present work we are concerned with the existence of normalized solutions to the following Schr\"odinger-Poisson System $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u + \mu (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ }…

We consider the semilinear electromagnetic Schr\"{o}dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(\Omega,\mathbb{C}), where $\Omega=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq…

In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as…

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 establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In…

The aim of this work is the study of the existence of normalized solutions to the nonlinear Schr\"odinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-\Delta u =\lambda…

In this paper we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\…

We consider the focusing mass supercritical nonlinear Schr\"odinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}\Delta u+\frac{1}{2}V(x)u-|u|^{p-1}u+L_{\Omega}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*}…

In the present paper, we study the normalized solutions for the following quasilinear Schr\"odinger equations: $$-\Delta u-u\Delta u^2+\lambda u=|u|^{p-2}u \quad \text{in}~\mathbb R^N,$$ with prescribed mass $$\int_{\mathbb R^N} u^2=a^2.$$…

Via a Lyapunov-Schmidt reduction, we obtain multiple semiclassical solutions to a class of fractional nonlinear Schr\"odinger equations. Precisely, we consider \begin{equation*} \varepsilon^{2s}(-\Delta)^{s}u+u+V(x)u=|u|^{p-1}u,\quad u\in…

We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 3$,} \] having…

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -\Delta u+ \lambda V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in…

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…

We investigate the existence of solutions to the fractional nonlinear Schr\"{o}dinger equation $(-\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly…

We consider the following nonlinear Schrodinger equation [{l} \Delta u-(1+\delta V)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $ f(u)$ is a superlinear nonlinearity satisfying…