Ground states of nonlinear Schrödinger systems

Chang, Jinyong; Liu, Zhaoli

doi:10.1090/S0002-9939-09-10090-4

Ground states of nonlinear Schrödinger systems
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by Jinyong Chang and Zhaoli Liu PDF

Proc. Amer. Math. Soc. 138 (2010), 687-693 Request permission

Abstract:

This paper concerns the existence of positive radial ground states of the time-independent Schrödinger system \begin{equation*} \left \{\begin {array}{ll} -\Delta {u_1}+\lambda _1u_1=\mu _1u_1^3+\beta {u_2^2u_1}, \quad &\text {in}\ \mathbb R^n,\\ -\Delta {u_2}+\lambda _2{u_2}=\mu _2u_2^3+\beta {u_1^2u_2}, \quad &\text {in}\ \mathbb R^n,\\ u_1(x)\to 0,\ \ u_2(x)\to 0,\ \ &\text {as}\ |x|\to \infty , \end{array}\right . \end{equation*} where $n=1,2,3$, $\lambda _j>0$ and $\mu _j>0$ for $j=1,2$, and $\beta >0$. A result from Sirakov, Comm. Math. Phys. 271 (2007), 199-221, is improved.

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