## 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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