Variational properties of the solutions of semilinear equations under nonresonance conditions


Abstract. The paper deals with weak solutions of the semilinear operator equation Au cu =J(u) in a Hilbert space, where Ais a positively defined linear operator, Jis a C1functional and cis not an eigenvalue of A. Under some assumptions on J, if Eis the energy functional of the equation and clies between two eigenvalues λk and λk+1,then for any solution uof the equation, E(u)E(u+w) for every element worthogonal on the first keigenvectors of A. The proof is based on the application of Ekeland’s variational principle to a suitable modified functional, and differs essentially from the proof of the particular case when c= 0.The theory is applicable to elliptic problems


Angela Budescu
Babes-Bolyai University, Department of Mathematics, 400084 Cluj, Romania

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


Semilinear operator equation; fixed point; critical point; eigenvalues; nonresonance;  minimizer; Ekeland’s variational principle; elliptic problem.

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A. Budescu, R. Precup, Variational properties of the solutions of semilinear equations under nonresonance conditions, J. Nonlinear Convex Anal. 17 (2016), 1517-1530.


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Journal of Nonlinear and Convex Analysis

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Yokohama Publishers

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