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Schrödinger systems with quadratic interactions. (English) Zbl 1437.35276
Let $$\lambda_1,\ \lambda_2, \ \mu_1, \mu_2$$ and $$\beta$$ be positive constants, and $$N\in [2,5]$$. The following type of Schrödinger system is studied in this paper: $(P): \ -\Delta u + \lambda_1 u = \mu_1 |u|u + \beta uv, \ -\Delta v + \lambda_2 v = \mu_2 |v|v + \frac{\beta}{2}u^2, \ x\in \mathbb{R}^N.$ Under some suitable conditions on $$\lambda_1,\ \lambda_2, \ \mu_1, \mu_2$$ and $$\beta$$, by variational methods, the existence of a positive ground state solution and the nonexistence of positive vector solutions are proved. Also the asymptotic behavior of ground state solutions is discussed for both $$\beta \rightarrow +\infty$$ and $$\beta \rightarrow 0$$. Moreover, since the associated variational functional is only partially symmetric (it is symmetric only with respect to the first component $$u$$ but not with respect to $$v$$), the usual symmetric critical point theorem of minimax type cannot be applied directly. The authors develop some new techniques for treating these kind of variational problems, upon which multiple bound state vector solutions are obtained.
##### MSC:
 35J57 Boundary value problems for second-order elliptic systems 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35J50 Variational methods for elliptic systems
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