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Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem. (English) Zbl 1207.35129

Summary: We study the following system of Maxwell-Schrödinger equations \[ \Delta u - u - \delta u\psi + f(u) = 0, \quad \Delta\psi + u^2 = 0 \text{ in } \mathbb R^N, \]
\[ u,\psi > 0, \quad u,\psi \to 0 \text{ as } | x| \to + \infty, \] where \(\delta > 0\), \(u\), \(\psi : \mathbb R^N \to \mathbb R\), \(f : \mathbb R \to \mathbb R\), \(N \geq 3\). We prove that the set of solutions has a rich structure: more precisely for any integer \(K\) there exists \(\delta_{K} > 0\) such that, for \(0 < \delta < \delta_{K}\), the system has a solution \((u_{\delta},\psi_{\delta})\) with the property that \(u_{\delta}\) has \(K\) spikes centered at the points \(Q_{1}^\delta,\ldots, Q_K^\delta\). Furthermore, setting \(l_\delta = \min_{i \not= j} | Q_i^\delta -Q_j^\delta|\), then, as \(\delta \rightarrow 0\), \((\frac{1}{l_\delta} Q_1^\delta,\ldots, \frac{1}{l_\delta} Q_K^\delta)\) approaches an optimal configuration for the following maximization problem: \[ \max\left\{ \sum_{i\neq j} \frac{1}{| Q_i-Q_j| ^{N-2}}\, (Q_i,\ldots, Q_K) \in \mathbb R^{NK},\, | Q_i-Q_j| \geq 1 \text{ for }i\neq j\right\}. \]

MSC:

35J57 Boundary value problems for second-order elliptic systems
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35J60 Nonlinear elliptic equations
35Q60 PDEs in connection with optics and electromagnetic theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
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