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- Title
Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem.
- Authors
D'Aprile, Teresa; Juncheng Wei
- Abstract
We study the following system of Maxwell-Schrödinger equations 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, ψδ) 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:
- Subjects
SCHRODINGER equation; DIFFERENTIAL equations; PARTIAL differential equations; EQUATIONS; WAVE mechanics; ALGEBRA
- Publication
Calculus of Variations & Partial Differential Equations, 2006, Vol 25, Issue 1, p105
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-005-0342-9