Bulletin of the
Korean Mathematical Society BKMS

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