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

## Abstract

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.

## Authors

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

## Keywords

Compactness; Hammerstein integral equation; Mountain pass theory.

## Paper cordinates

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

## PDF

##### Journal

Nonlinear Analysis

Elsevier

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0362-546X