## 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

### MSC:

 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

### Citations:

Zbl 0589.34013; Zbl 0615.35037