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Minimax principles for a class of lower semicontinuous functions and applications to nonlinear boundary value problems. (English) Zbl 0631.58002
Nonlinear functional analysis and its applications, Proc. NATO Adv. Study Inst., Maratea/Italy 1985, NATO ASI Ser., Ser. C 173, 393-399 (1986).
[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. $\psi\not\equiv \infty)$ and lower semicontinuous function, and $\phi \in C\sp 1(X,{\bbfR})$. It is also supposed that the following compactness condition is satisfied: If $(u\sb n)$ is a sequence such that $I(u\sb n)$ $(=\phi (u\sb n)+\psi (u\sb n))\to c\in {\bbfR}$ and $\phi '(u\sb n)+\partial \psi (u\sb n)\ni z\sb n$ where $z\sb n\to 0$, then $(u\sb n)$ 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 $I(0)=0$ and $\phi$, $\psi$ are even. Assume also that (i) there exists a subspace $X\sb 1$ of X of finite codimension, and numbers $\alpha,\rho >0$ such that $I\vert\sb{\partial B\sb{\rho}\cap X\sb 1}\ge \alpha$, (ii) there is a finite dimensional subspace $X\sb 2$ of X, dim $X\sb 2>co\dim X\sb 1$, such that I(u)$\to \infty$ as $\Vert u\Vert \to \infty$, $u\in X\sb 2$. Then I has at least dim $X\sb 2-co\dim X\sb 1$ distinct pairs of nontrivial critical points. Application to nonlinear boundary value problems are also given. Theorem 6. Let f(t) be an odd $C\sp 1$ function such that $f(0)=f'(0)=0$, f is nondecreasing and f’(t)$\to \infty$ as $\vert t\vert \to \infty$. If $\lambda >\lambda\sb k$, then the boundary value problem $(-\Delta)\sp mu+f(u)=\lambda (u)$ in $\Omega$, $u\in H\sb 0\sp m$ has at least k distinct pairs of nontrivial solutions u such that $uf(u)\in L\sp 1$. Here $\lambda\sb k$ is kth eigenvalue of $(-\Delta)\sp m$ in $H\sb 0\sp m$.
Reviewer: P.Kucment

58E05Abstract critical point theory
35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods