Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth. (English) Zbl 1352.35162
Summary: We consider the following nonlinear Schrödinger system with critical growth
\[
- \operatorname{\Delta} u_j = \lambda_j u_j + \sum_{i = 1}^k \beta_{i j} | u_i |^{\frac{2^\ast}{2}} | u_j |^{\frac{2^\ast}{2} - 2} u_j, \text{ in } \operatorname{\Omega}, u_j = 0, \text{ on } \partial \operatorname{\Omega}, j = 1, \cdots, k,
\]
where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(2^\ast = \frac{2 N}{N - 2}\), \(0 < \lambda_j < \lambda_1(\operatorname{\Omega})\), \(j = 1, \cdots, k\), \(\lambda_1(\operatorname{\Omega})\) is the first eigenvalue of with zero Dirichlet boundary condition. We consider the repulsive case, namely \(\beta_{j j} > 0\), \(j = 1, \cdots, k\), \(\beta_{i j} = \beta_{j i} \leq 0\), \(i \neq j\), \(i, j = 1, \cdots, k\). The existence of infinitely many sign-changing solutions as bound states is proved, provided \(N \geq 7\), by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 35A15 | Variational methods applied to PDEs |
| 35J50 | Variational methods for elliptic systems |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
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