Related papers: Nonlinear Fractional Schr\"odinger Equations coupl…

We look for solutions to the Schr\"odinger equation \[ -\Delta u + \lambda u = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = \rho^2$, with $N\ge2$. The behaviour of $g$ at the origin…

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schr\"odinger equation $$ \gamma \Delta ^2 u -\Delta u + \alpha u=|u|^{2 \sigma} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \,…

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 this paper, we study the following Schr\"odinger-Poisson system: $$ \left\{\aligned&-\Delta u+V_\lambda(x)u+K(x)\phi u=f(x,u)&\quad\text{in }\bbr^3,\\ &-\Delta\phi=K(x)u^2&\quad\text{in }\bbr^3,\\…

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0…

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…

In this paper, we consider the existence of solutions for the following fractional coupled Hartree-Fock type system \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u+V_1(x)u+\lambda_1u=\mu_1(I_{\alpha}\star…

We study the existence of ground and bound state solutions for a system of coupled Schr\"odinger equations with linear and nonlinear couplings in $\mathbb{R}^N$. By studying the limit system and using concentration compactness arguments, we…

In this article we are interested on the non-homogeneous fractional Schr\"odinger equation \begin{eqnarray}\label{eq00} &(-\Delta)^{\alpha}u(x) + V(x)u(x) = f(u) + h(x) \mbox{ in } \mathbb{R}^{n}. \end{eqnarray} By using mountain pass…

We consider a class of nonlinear Schr\"odinger equations with potential \[ i\partial_t u +\Delta u - Vu = \pm |u|^\alpha u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^3, \] where $\frac{4}{3}<\alpha<4$ and $V$ is a Kato-type potential. We…

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…

In this paper we investigate the existence of positive solutions to the following Schr\"odinger-Poisson-Slater system [c]{ll} - \Delta u+ u + \lambda\phi u=|u|^{p-2}u & \text{in} \Omega -\Delta\phi= u^{2} & \text{in} \Omega u=\phi=0 &…

The nonlinear Schr\"{o}dinger-Newton system \begin{equation*} \begin{cases} \Delta u- V(|x|)u + \Psi u=0, &~x\in\mathbb{R}^3,\\ \Delta \Psi+\frac12 u^2=0, &~x\in\mathbb{R}^3, \end{cases} \end{equation*} is a nonlinear system obtained by…

We consider the following nonlinear fractional Choquard equation, \begin{equation}\label{e:introduction} \begin{cases} (-\Delta)^{s} u + u = (1 + a(x))(I_\alpha \ast (|u|^{p}))|u|^{p - 2}u\quad\text{ in }\mathbb{R}^N,\\ u(x)\to 0\quad\text{…

In this paper, we consider the following weakly coupled nonlinear Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} -\epsilon^{2}\Delta u_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + \beta|u_1|^{p - 2}|u_2|^pu_1, & x\in…

In this paper, we are concerned with solutions to the following nonlinear Schr\"odinger equation with combined inhomogeneous nonlinearities, $$ -\Delta u + \lambda u= \mu |x|^{-b}|u|^{q-2} u + |x|^{-b}|u|^{p-2} u \quad \mbox{in} \,\, \R^N,…

In this paper, we introduce some new ideas to study Schrodinger equations in RN with power-type nonlinearities.

We consider the following system linearly coupled by nonlinear Schr\"odinger equations in $\R^3$ $$ \left\{\begin{array}{ll} -\Delta u_j+u_j=u^3_j-\va\sum\limits_{i\neq j}^N u_i,\{1cm}& x\in \R^3, \{0.2cm}\\ u_j\in H^1(\R^3),\quad…

In this paper we focus our attention on the following nonlinear fractional Schr\"odinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{N},…

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}…