On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces

In this paper we deal with nontrivial solvability in balls of Hammerstein integral equations in Hilbert spaces for nonlinearities of potential type. We use a variational approach based on variants of the mountain pass theorem which are due to Guo—Sun-Qi and Schechter. Our main contribution is a new technique to verify compactness conditions of Palais-Smale type. This technique combines the compactness criterium for countable sets in \(L^p\) with basic properties of the measures of noncompactness and integral inequalities.

Radu Precup
Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania

Compactness; Hammerstein integral equation; Mountain pass theory.

R. Precup, On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces, Nonlinear  Anal. 47 (2001), 1233-1244. http://dx.doi.org/10.1016/S0362-546X(01)00261-9

http://ictp.acad.ro/wp-content/uploads/2022/08/2001f-Precup-On-the-Palais-Smale-condition-for-Hammerstein-integral-equations-in-Hilbert-spaces-Nonlinear-Anal.-47-2001-1233-1244.pdf

http://dx.doi.org/10.1016/S0362-546X(01)00261-9

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