Related papers: Normalized solutions for the fractional NLS with m…

We study existence of solutions for the fractional problem \begin{equation*} (P_m) \quad \left \{ \begin{aligned} (-\Delta)^{s} u + \mu u &=g(u) & \; \text{in $\mathbb{R}^N$}, \cr \int_{\mathbb{R}^N} u^2 dx &= m, & \cr u \in…

Using a Fourier spectral method, we provide a detailed numerically investigation of dispersive Schr\"odinger type equations involving a fractional Laplacian. By an appropriate choice of the dispersive exponent, both mass and energy sub- and…

In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schr\"{o}dinger equation \begin{equation} i\partial_{t}\psi=(-\Delta)^{s}\psi-f(\psi), \qquad (0.1)\end{equation} where…

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 \ }…

In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -\Delta_{p}u+{\vert u\vert}^{p-2}u=f(u)-\mu u \quad…

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…

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.…

This paper focuses on the existence of multiple normalized solutions to Schr\"{o}dinger equations with general nonlinearities in bounded domains via variational methods. We first obtain two positive normalized solutions, one is a normalized…

In this paper we deal with the following nonlocal systems of fractional Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v)+\gamma H_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\\…

In this paper, we study the existence of ground state solutions for the nonlinear fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^su+V(x)u+\phi u=|u|^{p-1}u, & \hbox{in $\mathbb{R}^3$,}…

We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of…

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+\mu |u|^{q-2}u+f(u), \quad \quad \hbox{in…

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-\Delta)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and…

We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{l} -\Delta u_i+ \lambda_i u_i = \sum_{j=1}^K\beta_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2}…

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where…

In this paper, we study the existence and asymptotic properties of solutions to the following fractional Kirchhoff equation \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u=\lambda…

We study the existence of solutions of the following nonlinear Schr\"odinger equation \begin{equation*} -\Delta u + \Big(V(x)-\frac{\mu}{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where…

Let $\sigma\in(0,1)$ with $\sigma\neq\frac{1}{2}$. We investigate the fractional nonlinear Schr\"odinger equation in $\mathbb R^d$: $$i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-\Delta)^\sigma$ is the…

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 establish global well-posedness for the mass subcritical nonlinear fractional Schr\"odinger equation $$iu_t - (-\Delta)^\frac{\beta}{2} u+F(u)=0$$ having radial initial data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R^n)$ for $n…