相关论文: On Entire Solutions of an Elliptic System Modeling…

We study the one dimensional symmetry of entire solutions to an elliptic system arising in phase separation for Bose-Einstein condensates with multiple states. We prove that any monotone solution, with arbitrary algebraic growth at…

We prove the existence of entire solutions with exponential growth for the semilinear elliptic system [\begin{cases} -\Delta u = -u v^2 & \text{in $\R^N$} -\Delta v= -u^2 v & \text{in $\R^N$} u,v>0, \end{cases}] for every $N \ge 2$. Our…

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: \[ \begin{cases} -\Delta u = -u v^2 & \text{in $\R^N$}\\ -\Delta v= -u^2 v & \text{in $\R^N$}, \end{cases} \]…

We consider solutions of the competitive elliptic system \[ \left\{ \begin{array}{ll} -\Delta u_i = - \sum_{j \neq i} u_i u_j^2 & \text{in $\mathbb{R}^N$} \\ u_i >0 & \text{in $\mathbb{R}^N$} \end{array}\right. \qquad i=1,\dots,k. \] We are…

In this paper we study the one dimensional symmetry problem of entire solutions to the problem \[\Delta u=uv^2,\Delta v=vu^2,u,v>0 \text{in} \mathbb{R}^n,\] for all $n\geq 2$. We prove that, if a solution $(u,v)$ is a local minimizer and…

We consider strongly coupled competitive elliptic systems that arise in the study of two-component Bose-Einstein condensates. As the coupling parameter tends to infinity, solutions that remain uniformly bounded are known to converge to a…

For the system of semilinear elliptic equations \[ \Delta V_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in $\mathbb{R}^N$} \] we devise a new method to construct entire solutions. The method extends the existence results…

We consider a family of positive solutions to the system of $k$ components \[ -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in $\Omega$}, \] where $\Omega \subset \mathbb{R}^N$…

We study the inhomogeneous linear system which arises in the higher order asymptotic expansion of the wave functions of multi-component Bose-Einstein condensates, across the regular part of the interface, in the case of segregation.

The cubic nonlinear Schrodinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose--Einstein condensate in a lattice potential. A family of exact stationary solutions is…

The cubic nonlinear Schrodinger equation with a lattice potential is used to model a periodic dilute gas Bose-Einstein condensate. Both two- and three-dimensional condensates are considered, for atomic species with either repulsive or…

In this paper we consider a two component system of coupled non linear Schr\"odinger equations modeling the phase separation in the binary mixture of Bose-Einstein condensates and other related problems. Assuming the existence of solutions…

In this paper, we study the following two-component systems of nonlinear Schr\"odinger equations \begin{equation*} \left\{\aligned&\Delta u-(\lambda a(x)+a_0(x))u+\mu_1u^3+\beta v^2u=0\quad&\text{in }\bbr^3,\\ &\Delta v-(\lambda…

We consider the inner limit system describing the phase separation in two-component Bose-Einstein condensates linearized around the one-dimensional solution in an infinite strip with zero and periodic boundary conditions, and obtain optimal…

We consider the following elliptic system with fractional laplacian $$ -(-\Delta)^su=uv^2,\ \ -(-\Delta)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n,$$ where $s\in(0,1)$ and $(-\Delta)^s$ is the $s$-Lapalcian. We first prove that all positive…

We study properties of the semilinear elliptic equation $\Delta u = 1/u$ on domains in $R^n$, with an eye toward nonnegative singular solutions as limits of positive smooth solutions. We prove the nonexistence of such solutions in low…

The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of…

We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \[ \left\{ \begin{tabular} [c]{l}% $-\Delta u=\mu_{1}|u|^{{2}^{\ast}-2}u+\lambda\alpha|u|^{\alpha-2}|v|^{\beta }u,$\\ $-\Delta…

The time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r interaction is investigated with both a variational approach and numerically exact calculations. We show that these condensates exhibit…

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following…