Related papers: Normalized bound state solutions of fractional Sch…

In this paper, we study the following fractional Schr\"{o}dinger equation with prescribed mass \begin{equation*} \left\{ \begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\quad\text{in $\mathbb{R}^{N}$},\\…

In this paper, we find normalized solutions to the following Schr\"{o}dinger equation \begin{equation}\notag \begin{aligned} &-\Delta u-\frac{\mu}{|x|^2}h(x)u+\lambda u =f(u)\quad\text{in}\quad\mathbb{R}^{N},\\ & u>0,\quad…

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…

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of Schr\"{o}dinger equations with Hardy potentials: \begin{equation*} \left\{ \begin{aligned}…

In this paper we study the following fractional Choquard equation with mixed nonlinearities: \[ \left\{ \begin{array}{l} (-\Delta)^s u = \lambda u + \alpha \left( I_\mu * |u|^q \right) |u|^{q-2} u + \left( I_\mu * |u|^p \right) |u|^{p-2} u,…

In this paper, we are concerned with the following fractional $N/s$-Laplacian Choquard equation \begin{align*} \begin{cases} (-\Delta)^s_{N/s}u=\lambda |u|^{\frac{N}{s}-2}u +(I_\mu*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}^N,…

We consider the following coupled fractional Schr\"{o}dinger system: \begin{equation*} \left\{ \begin{aligned} &(-\Delta)^su+\lambda_1u=\mu_1|u|^{2p-2}u+\beta|v|^p|u|^{p-2}u\\ &(-\Delta)^sv+\lambda_2v=\mu_2|v|^{2p-2}v+\beta|u|^p|v|^{p-2}v\\…

This paper investigates the existence of normalized solutions to the nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(x) u=\lambda u+f(x)\left(I_\alpha *\left(f|u|^q\right)\right)|u|^{q-2} u+g(x)\left(I_\alpha…

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 fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…

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 fractional Schr\"{o}dinger-Poisson equation \begin{equation*} \ \left\{\begin{aligned} &(-\Delta)^{s}u+V(x)u+K(x)\phi u=|u|^{2^{\ast}_{s}-2}u, &\mbox{in} \ \mathbb{R}^{3},\\…

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…

We revisit the following fractional Schr\"{o}dinger equation \begin{align}\label{1a} \varepsilon^{2s}(-\Delta)^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \mathrm{in}\ \R^N, \end{align} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes…

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,\\…

In this paper, we study the following fractional Schr\"odinger equation: \[ \left\{\begin{gathered} {(- \Delta)^s}u + mu = f(u){\text{in}}{\mathbb{R}^N}, \hfill u \in {H^s}({\mathbb{R}^N}),{\text{}}u > 0{\text{on}}{\mathbb{R}^N}, \hfill \\…

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1),…

We study the following fractional Schr\"{o}dinger equation \begin{equation*}\label{eq0.1} \epsilon^{2s}(-\Delta)^s u + V(x)u = |u|^{p - 2}u, \,\,x\in\,\,\mathbb{R}^N, \end{equation*} where $s\in (0,\,1)$, $N>2s$, $p>1$ is subcritical and…

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schr\"odinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1+ \lambda_1 u_1= \mu_1 |u_1|^{2p-2}u_1+\beta |u_2|^{p}…

We look for normalized solutions to the nonlinear Schr\"{o}dinger equation with mixed fractional Laplacians and combined nonlinearities $$ \left\{\begin{array}{ll} (-\Delta)^{s_{1}} u+(-\Delta)^{s_{2}} u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u…