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Standing waves of some coupled nonlinear Schrödinger equations. (English) Zbl 1130.34014

This paper under review mainly deals with the system
\[ \begin{cases} -\Delta u_1+ \lambda_1u_1=\mu_1u_1^3+\beta u_1u_2^2,\\ -\Delta u_2+\lambda_2u_2=\mu_2u_2^3+ \beta u^2_1u_2\end{cases} \]
where the solution satisfies \(u_1,u_2\in W^{1,2} (\mathbb{R}^n)\), with \(n=2\) or 3, \(\lambda_1,\lambda_2,\mu_1,\mu_2>0\), and \(\beta\in \mathbb{R}\). For \(n=1\) and \(\mu_1=\mu_2\), this system describes the so-called standing waves of a system of two coupled nonlinear Schrödinger type equations. The authors use a natural energy, and denote the non-trivial critical points of the energy nontrivial bound states of the system. A positive bound state is called a ground state if its energy is minimal among all nontrivial bound states. In addition, they introduce two constants \(\Lambda\leq \Lambda'\) in terms of the unique positive radial solution \(U\) of \(-\Delta u+u=u^3\), which has been proven to exist. Note that the system has two known solutions \((u_1,u_2)=(U_1,0)\) and \((0,U_2)\), where \(U_j(x)=\sqrt{\lambda_j/\mu_j}U(\sqrt {\lambda_j}x)\) for \(j=1\) and 2. The main results of the paper are: if \(\beta< \Lambda\), then the system has a radial bound state other than the two known solutions; if \(\beta> \Lambda'\), then the system has a radial ground state. Similar systems on more than two functions are also discussed.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
34B15 Nonlinear boundary value problems for ordinary differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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