*(English)*Zbl 0631.58002

[For the entire collection see Zbl 0583.00019.]

Let X be a real Banach space, $\psi $ : $X\to (-\infty ,\infty ]$ a convex, proper (i.e. $\psi \neg \equiv \infty )$ and lower semicontinuous function, and $\varphi \in {C}^{1}(X,\mathbb{R})$. It is also supposed that the following compactness condition is satisfied: If $\left({u}_{n}\right)$ is a sequence such that $I\left({u}_{n}\right)$ $(=\varphi \left({u}_{n}\right)+\psi \left({u}_{n}\right))\to c\in \mathbb{R}$ and ${\varphi}^{\text{'}}\left({u}_{n}\right)+\partial \psi \left({u}_{n}\right)\ni {z}_{n}$ where ${z}_{n}\to 0$, then $\left({u}_{n}\right)$ possesses a convergent subsequence. Here $\partial \psi $ is the subdifferential of $\psi $.

The author gives the ideas of proofs for some criteria for existence of critical points. For example: Theorem 3. Let $I\left(0\right)=0$ and $\varphi $, $\psi $ are even. Assume also that (i) there exists a subspace ${X}_{1}$ of X of finite codimension, and numbers $\alpha ,\rho >0$ such that ${I|}_{\partial {B}_{\rho}\cap {X}_{1}}\ge \alpha $, (ii) there is a finite dimensional subspace ${X}_{2}$ of X, dim ${X}_{2}>codim{X}_{1}$, such that I(u)$\to \infty $ as $\parallel u\parallel \to \infty $, $u\in {X}_{2}$. Then I has at least dim ${X}_{2}-codim{X}_{1}$ distinct pairs of nontrivial critical points.

Application to nonlinear boundary value problems are also given. Theorem 6. Let f(t) be an odd ${C}^{1}$ function such that $f\left(0\right)={f}^{\text{'}}\left(0\right)=0$, f is nondecreasing and fâ€™(t)$\to \infty $ as $\left|t\right|\to \infty $. If $\lambda >{\lambda}_{k}$, then the boundary value problem ${(-{\Delta})}^{m}u+f\left(u\right)=\lambda \left(u\right)$ in ${\Omega}$, $u\in {H}_{0}^{m}$ has at least k distinct pairs of nontrivial solutions u such that $uf\left(u\right)\in {L}^{1}$. Here ${\lambda}_{k}$ is kth eigenvalue of ${(-{\Delta})}^{m}$ in ${H}_{0}^{m}$.