We study the system of Maxwell-Schro ̈dinger equations ∆u−u−δuψ+f(u)=0, ∆ψ+u2 =0 in RN, where δ>0, u,ψ:RN→R, u,ψ>0, u,ψ→0 as |x|→+∞, and f:R→R,N≥3. We prove that the set of solutions has a rich structure; more precisely, for any integer K there exists δK>0 such that, for 0<δ<δK, the system has a solution (uδ,ψδ) with the property that uδ has K spikes centered at the points Q_δ1,...,Q_δK. Furthermore, for lδ = mini̸=j |Q_δi −Q_δj|, then, as δ → 0, ((1/lδ) Q_δ1,...,(1/lδ) Q_δK) approaches an optimal configuration for the following maximization problem: max{Σi≠j 1/|Q_i-Q_j|^{N-2} |(Qi,...,QK)∈R^NK,|Q_i-Q_j|≥1 for i≠j}
D'Aprile, T.c., Wei, J. (2006). Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 25(1), 105-137 [10.1007/s00526-005-0342-9].
Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem
D'APRILE, TERESA CARMEN;
2006-01-01
Abstract
We study the system of Maxwell-Schro ̈dinger equations ∆u−u−δuψ+f(u)=0, ∆ψ+u2 =0 in RN, where δ>0, u,ψ:RN→R, u,ψ>0, u,ψ→0 as |x|→+∞, and f:R→R,N≥3. We prove that the set of solutions has a rich structure; more precisely, for any integer K there exists δK>0 such that, for 0<δ<δK, the system has a solution (uδ,ψδ) with the property that uδ has K spikes centered at the points Q_δ1,...,Q_δK. Furthermore, for lδ = mini̸=j |Q_δi −Q_δj|, then, as δ → 0, ((1/lδ) Q_δ1,...,(1/lδ) Q_δK) approaches an optimal configuration for the following maximization problem: max{Σi≠j 1/|Q_i-Q_j|^{N-2} |(Qi,...,QK)∈R^NK,|Q_i-Q_j|≥1 for i≠j}I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
