Limit relative category theory applied to the critical point theory
Author(s)
Tacksun Jung and Q-Heung Choi
MSC
Abstract
Let $H$ be a Hilbert space which is the direct sum
of five closed subspaces $X_{0}$, $X_{1}$, $X_{2}$, $X_{3}$ and
$X_{4}$ with $X_{1}$, $X_{2}$, $X_{3}$ of finite dimension. Let
$J$ be a $C^{1,1}$ functional defined on $H$ with $J(0)=0$. We
show the existence of at least four nontrivial critical points
when the sublevels of $J$ (the torus with three holes and sphere)
link and the functional $J$ satisfies sup-inf variational
inequality on the linking subspaces, and the functional $J$
satisfies $(P.S.)^{*}_{c}$ condition and $f|_{X_{0}\oplus X_{4}}$
has no critical point with level $c$. For the proof of main
theorem we use the nonsmooth version of the classical deformation
lemma and the limit relative category theory.
Keyword
$C^{1,1}$ functional, nonsmooth version classical deformation lemma, limit relative category theory, critical point theory, manifold with boundary, $(P.S.)^{*}_{c}$ condition
