Semi-coercive monotone variational problems. (English) Zbl 0647.49007

Let H be a real Hilbert space, \(H_ 1\) a closed subspace and \(H=H_ 0\oplus H_ 1\), where \(H_ 0=H^{\perp}_ 1\). Let \(a: H\to {\mathbb{R}}\cup \{+\infty \}\) and \(g: H\to {\mathbb{R}}\) be convex lower semi- continuous functions, a proper and \(H_ 1\)-coercive and \(g(u_ 0)\to +\infty\), as \(\| u_ 0\| \to +\infty\) in \(H_ 0\). Then it is shown that for \(f=a+g\) the set of critical points is non-empty and bounded. Applications to some ordinary and elliptic boundary value problems are given.
The results generalize and complete earlier results of R. Kannan, J. J. Nieto and M. B. Ray [J. Math. Anal. Appl. 105, 1-11 (1985; Zbl 0589.34013)], and J. J. Nieto [Boll. Unione Mat. Ital., VI Ser., A 5, 205-210 (1986; Zbl 0615.35037)] and the author.
Reviewer: V.Mustonen


49J45 Methods involving semicontinuity and convergence; relaxation
49J27 Existence theories for problems in abstract spaces
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
35J20 Variational methods for second-order elliptic equations
46C99 Inner product spaces and their generalizations, Hilbert spaces