## Abstract

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

## Authors

Angela Budescu

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

**Radu Precup**

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

## Keywords

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

## Paper coordinates

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

## About this paper

##### Journal

Journal of Nonlinear and Convex Analysis

##### Publisher Name

Yokohama Publishers

##### Print ISSN

1345-4773

##### Online ISSN

1880-5221

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