Multiple solutions for critical elliptic systems via penalization method. (English) Zbl 1240.35144

Alves, Claudianor O.; Figueiredo, Giovany M.; Furtado, Marcelo F.

Differ. Integral Equ. 23, No. 7-8, 703-723 (2010).

Summary: We consider the system \[ -\varepsilon ^2\Delta u+W(x)u=Q_{u}(u,v)+\frac {1}{2^{*}}K_{u}(u,v), \]
\[ -\varepsilon ^2\Delta v+V(x)v=Q_{v}(u,v)+\frac {1}{2^{*}}K_{v}(u,v) \] in \(\mathbb {R}^{N}\) with positive initial data, where \(2^{*}=2N/(N - 2)\), \(N\geq 3\), \(\varepsilon >0\) is a parameter, \(W\) and \(V\) are positive potentials, and \(Q\) and \(K\) are homogeneous function with \(K\) having critical growth. We relate the number of solutions to the topology of the set, where \(W\) and \(V\) attain their minimum values. We apply Ljusternik-Schnirelmann theory in the proof.

MSC:

35J47	Second-order elliptic systems
35J50	Variational methods for elliptic systems
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

critical elliptic system; multiple solution; Ljusternik-Schnirelmann theory