Variational properties of the solutions of semilinear equations under nonresonance conditions


The paper deals with weak solutions of the semilinear operator equation \(Au-cu=J^{\prime}\left( u\right))\ in a Hilbert space, where \(A)\ is a positively defined linear operator, \(J)\ is a \(C^{1})\ functional and \(c)\ is not an eigenvalue of \(A)\. Under some assumptions of \(J)\, if \(E)\ is the energy functional of the equation and \(c)\ lies between two eigenvalues \(\leftthreetimes_{k})\ and \(\leftthreetimes_{k+1})\, then for any solution \(u)\ of the equation, \(E\left( u\right) \leq E\left( u+\omega\right))\ for every element \(\omega)\ orthogonal on the first \(k)\ eigenvectors of \(A)\. The proof is based on the application of Ekeland’s variational principle to a suitable modified functional, and differs essentially from the prooof 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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