Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem

Abstract

We study the following system of Maxwell-Schrödinger equations

$$ \Delta u - u - \delta u \psi+ f(u)=0, \quad \Delta \psi + u^2 = 0 \mbox{in} {\mathbb R}^N , u, \;\psi > 0, \quad u, \;\psi \to 0 \ \mbox{as} \ |x| \to + \infty, $$

where δ > 0, u, ψ : $\psi: {\mathbb R}^N \to {\mathbb R}$, f : ${\mathbb R} \to {\mathbb R}$, N ≥ 3. We prove that the set of solutions has a rich structure: more precisely for any integer K there exists δ _K > 0 such that, for 0 < δ < δ _K , the system has a solution (u _δ, ψ_δ) with the property that u _δ 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 δ → 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:

$$ \max\bigg\{\sum_{{i\neq j}}\frac{1}{|Q_i-Q_j|^{N-2}}\,\Big|\, (Q_i,\ldots, Q_K)\in {\mathbb R}^{NK},\, |Q_i-Q_j|\geq 1\hbox{ for }i\neq j\bigg\}. $$