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Critical point theory of Ljusternik-Schnirelmann type and applications to partial differential equations. (English) Zbl 0701.58016
Minimax results of Lusternik-Schnirelman type and applications, Proc. NATO ASI Var. Methods Nonlinear Probl., Montréal/Can. 1986, Sémin. Math. Supér., Sémin. Sci. OTAN (NATO Adv. Study Inst.) 107, 35-96 (1989).
[For the entire collection see Zbl 0694.00016.]
The bulk of this paper was presented at a seminar in 1986 and is largely of a review character. It treats various aspects of Ljusternik- Schnirelman critical point theory on differentiable manifolds and Banach space.
Whereas the first sections (2-5) present a variety of standard results in this area, the latter two sections present newer results some of which are due to the author. Included here are some highly interesting applications to boundary value problems for systems in bounded domains of $${\mathbb{R}}^ N:$$ $-\Delta u=F'_ s(u,v)\text{ in } D,\quad u|_{bd(D)}=0;\quad \Delta v=F'_ t(u,v)\text{ in } D,\quad v|_{bd(D)}=0,$ with D bounded domain in $${\mathbb{R}}^ N$$ and $$F=F(s,t)$$ in $$C^ 1({\mathbb{R}}^ 2,{\mathbb{R}})$$ (and $$bd(D)=boundary$$ of D).