Related papers: On least energy solutions for a nonlinear Schr\"od…

We study the following gradient elliptic system with Neumann boundary conditions \begin{equation*} -\Delta u + \lambda_1 u = u^3 + \beta uv^2, \ -\Delta v + \lambda_2 v = v^3 + \beta u^2 v \ \text{in } \Omega,\qquad \frac{\partial…

In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schr\"odinger system \begin{align*} \begin{gathered} -\Delta u + \, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q…

Let $\Omega\subset \mathbb{R}^{N}$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schr\"{o}dinger system with $d\geq 2$ equations \begin{equation*} -\Delta…

In the present paper, we consider the coupled Schr\"{o}dinger systems with critical exponent: \begin{equation*} \begin{cases} -\Delta u_i+\lambda_{i}u_i=\sum\limits_{j=1}^{d} \beta_{ij}|u_j|^{3}|u_i|u_i \quad ~\text{ in } \Omega,\\ u_i \in…

In this paper we deal with the following weakly coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} - \Delta_\alpha u + \omega u = |u|^2 u + \beta u |v|^2&\quad \mathrm{in}\ \mathbb{R}^2,\\ - \Delta v + \tilde{\omega} v =…

In this paper we investigate the existence of solutions to the following Schr\"{o}dinger system in the critical case \begin{equation*} -\Delta u_{i}+\lambda_{i}u_{i}=u_{i}\sum_{j = 1}^{d}\beta_{ij}u_{j}^{2} \text{ in } \Omega, \quad u_i=0…

We are concerned with the following coupled Schr\"{o}dinger system with Hardy potential in the critical case \begin{equation*} \begin{cases} -\Delta u_{i}-\frac{\lambda_{i}}{|x|^2}u_{i}=|u_i|^{2^*-2}u_i+\sum_{j\neq…

We prove a conjecture which was recently formulated by Maia, Montefusco, Pellacci saying that minimal energy solutions of the saturated nonlinear Schr\"odinger system \begin{align*} - \Delta u + \lambda_1 u &= \frac{\alpha u(\alpha…

In this paper we focus on existence and symmetry properties of solutions to the cubic Schr\"odinger system \[ -\Delta u_i +\lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i \quad \text{in $\Omega \subset \mathbb{R}^N$},\qquad i=1,\dots d \]…

Consider the following nonlinear Schr\"odinger--Bopp--Podolsky system in $\mathbb{R}^3$: \[ \begin{cases} - \Delta v + v + \phi v = v |v|^{p - 2}; \\ \beta^2 \Delta^2 \phi - \Delta \phi = 4 \pi v^2, \end{cases} \] where $\beta > 0$ and $3 <…

In this paper, we study the following $k$-coupled nonlinear Schr\"odinger system with Sobolev critical exponent: \begin{equation*} \left\{ \begin{aligned} -\Delta u_i & +\lambda_iu_i =\mu_i u_i^{2^*-1}+\sum_{j=1,j\ne i}^{k} \beta_{ij}…

In this paper, we study the following fractional nonlinear Schr\"odinger system $$ \left\{% \begin{array}{ll} (-\Delta)^s u +u=\mu_1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u,~~x\in \R^N,\vspace{2mm}\\ (-\Delta)^s v +v=\mu_2 |v|^{2p-2}v+\beta…

We study a nonlinear Schr\"odinger system with three-wave interaction: \begin{equation*} \left\{\begin{aligned} & - \Delta u_1 = f_1(u_1) + \alpha u_2u_3 \quad \text{ in } \R^N, & - \Delta u_2 = f_2(u_2) + \alpha u_3u_1 \quad \text{ in }…

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset…

Consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb{R}^3$ under an $L^2$-norm constraint, \[ \begin{cases} -\Delta u + \omega u + \phi u = u|u|^{p-2},\newline -\Delta \phi + a^2\Delta^2\phi=4\pi u^2,\newline…

We look for least energy solutions to the cooperative systems of coupled Schr\"odinger equations \begin{equation*} \begin{cases} -\Delta u_i + \lambda_i u_i = \partial_iG(u)\quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, u_i \in…

In this paper we consider the following Schr\"odinger-Poisson system in the whole $\mathbb R^{3}$, \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+ \lambda \phi u=f(u) &\text{ in } \mathbb R^3, -\Delta \phi= u^2 &\text{ in } \mathbb…

This is the first of two papers which study asymptotic behavior of minimal energy solutions to the fractional Lane-Emden system in a smooth bounded domain $\Omega$ \[(-\Delta)^s u = v^p, \quad (-\Delta)^s v = u^q \text{ in } \Omega \quad…

In this paper, we are concerned with the following Schr\"{o}dinger-Poisson system with critical nonlinearity and critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality \begin{equation}\begin{cases} -\Delta u+u+\lambda\phi…

We find positive non-radial solutions for a system of Schr\"odinger equations in a weak fully attractive or repulsive regime in presence of an external radial trapping potential that exhibits a maximum or a minimum at infinity.