Standing waves in the Maxwell-Schr�dinger equation and an optimal configuration problem

D'Aprile, Teresa; Juncheng Wei
January 2006
Calculus of Variations & Partial Differential Equations;Jan2006, Vol. 25 Issue 1, p105
Academic Journal
We study the following system of Maxwell-Schr�dinger equations where d > 0, u, ? : $$\psi: {\mathbb R}^N \to {\mathbb R}$$ , 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 d K > 0 such that, for 0 < d < d K , the system has a solution ( u d, ?d) with the property that u d 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 d ? 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:


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