Limit relative category theory applied to the critical point theory
Bull. Korean Math. Soc. 2009 Vol. 46, No. 2, 311-319
https://doi.org/10.4134/BKMS.2009.46.2.311
Printed March 1, 2009
Tacksun Jung and Q-Heung Choi
Kunsan National University and Inha University
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.
Keywords : $C^{1,1}$ functional, nonsmooth version classical deformation lemma, limit relative category theory, critical point theory, manifold with boundary, $(P.S.)^{*}_{c}$ condition
MSC numbers : 35R99
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