Related papers: Least energy positive soultions for $d$-coupled Sc…

We consider the following Scr\"odinger system $$\begin{cases}\displaystyle i\partial_t u + \Delta u +(|u|^2+\beta |v|^2) u= 0, \\ \displaystyle i\partial_t v + \Delta v +(|v|^2+\beta |u|^2) v = 0,\end{cases}$$ with initial data $(u_0,v_0)…

We study the following Lane-Emden system \[ -\Delta u=|v|^{q-1}v \quad \text{ in } \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \text{ in } \Omega, \qquad u_\nu=v_\nu=0 \quad \text{ on } \partial \Omega, \] with $\Omega$ a bounded regular…

In this paper, we are concerned with the existence of the least energy sign-changing solutions for the following fractional Schr\"{o}dinger-Poisson system: \begin{align*} \left\{ \begin{aligned} &(-\Delta)^{s} u+V(x)u+\lambda\phi(x)u=f(x,…

In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -\Delta u & =\lambda |u|^{p-2}u…

We study the following problem \[ \begin{cases} -\Delta u = \lambda u + u^{2^*-2} v & \hbox{in} \Omega,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} & \hbox{in} \Omega,\\ u> 0,v> 0 & \hbox{in} \Omega,\\ u=v=0 & \hbox{on} \partial \Omega,…

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…

We study a nonlinear Schr\"{o}dinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- \Delta u+ u + \lambda^2 \left(\frac{1}{\omega|x|^{N-2}}\star \rho u^2\right) \rho(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \]…

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 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 are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…

The Lane-Emden system is written as \begin{equation*} \begin{cases} -\Delta u = v^p &\text{in } \Omega,\\ -\Delta v = u^q &\text{in } \Omega,\\ u, v > 0 &\text{in } \Omega,\\ u = v = 0 &\text{on } \partial \Omega \end{cases} \end{equation*}…

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \label{0.1} {& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \text{in} \Omega, & u=0 \qquad \text{on}…

We study the pure Neumann Lane-Emden problem in a bounded domain \[ -\Delta u = |u|^{p-1} u \text{ in }\Omega, \qquad \partial_\nu u=0 \text{ on }\partial \Omega, \] in the subcritical, critical, and supercritical regimes. We show existence…

We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v,…

We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schr\"odinger equation $$ \begin{array}{c} -\Delta u +V(x)u=\lambda a(x)u^{-\gamma}+b(x)u^{p}~\mbox{in}~ \mathbb{R}^{N}, \end{array}…

In this paper, we study the ground state solutions of the following coupled nonlinear Schr\"odinger system (P) $-\Delta u_1-\tau_1 u_1 =\mu_1u_1^3+\beta u_1u_2^2$, $ -\Delta u_2-\tau_2 u_2 =\mu_2u_2^3+\beta u_1^2u_2$ in $\Omega$,…

In this paper, we consider the following nonlinear Schr\"odinger system in $R^3$: \begin{align*} -\Delta u_j +P_j(x) u=\mu_j u_j^3+\sum\limits_{i=1,i\neq j}^N\beta_{ij}u_i^2u_j, \end{align*} where $N\geq3$, $P_j$ are nonnegative radial…

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: {displaymath} {cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega, -\Delta u_2 +\la_2 u_2…

In this paper we deal with the cubic Schr\"odinger system $ -\Delta u_i = \sum_{j=1}^n \beta_{ij}u_j^2 u_i$, $u_1,\dots,u_n \geq 0$ in $\mathbb{R}^N (N\leq 3)$, where $\beta=(\beta_{i,j})_{ij}$ is a symmetric matrix with real coefficients…

In this paper, we look for solutions to the following coupled Schr\"{o}dinger system \begin{equation*} \begin{cases} -\Delta u+\lambda_{1}u=\alpha_{1}|u|^{p-2}u+\mu_{1}u^{3}+\rho v^{2}u & \text{in} \ \ \mathbb{R}^{N}, -\Delta…