In this note some well-known existence and multiplicity results of nontrivial solutions for scalar Hammerstein equations ,  are extended to equations in Hilbert spaces. The tools are a mountain pass theorem on closed convex substes of a Hilbert space due to Guo-Sun-Qi  and a new technique of checking the Palais-Smale compactness condition which was first presented in . The results compplement those established in .
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Hammerstein integral equation; compactness; critical point theory.
R. Precup, Nontrivial solvability of Hammerstein integral equations in Hilbert spaces, Seminaire de la Theorie de la Mielleure Approximation Convexite et Optimisation, Cluj-Napoca, 26 octombre – 29 octobre, 2000, pp. 255-265.
Seminaire de la Theorie de la Meilleure Approximation, Convexite et Optimisation
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 D. Guo, J. Sun, G. Qi, Some extensions of the mountain pass lemma, Differential Integral Equaitons 1 (1988), 351-358.
 M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral equations, Pergamon Press, Oxford, 1964.
 R. Precup, Nonlinear Integral Equations (Romanian), Babes-Bolyai Univ. Cluj, 1993.
 R. Precup, On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces, to appear.
 P.H. Rabinowits, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer., Math. Soc., Providence, 1986.