Variational properties of the solutions of semilinear equations under nonresonance conditions

Radu Precup
not-at-ICTP

Abstract

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.

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.

PDF

https://www.researchgate.net/publication/309315681

http://www.yokohamapublishers.jp/online2/jncav21-7.html

About this paper

Journal

Journal of Nonlinear and Convex Analysis

Publisher Name

Yokohama Publishers

DOI

https://www.researchgate.net/publication/309315681

Print ISSN

1345-4773

Online ISSN

1880-5221

google scholar link

[1] H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci.7(1980), 539–603.
[2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.
[3] A. Budescu, Semilinear operator equations and systems with potential-type nonlinearities, Studia Univ. Babes-Bolyai Math. 59 (2014), 199–212.
[4] J. Mawhin and J. R. Ward Jr, Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Anal. 5(1981), 677–684.
[5] S. G. Michlin, Linear Partial Differential Equations (Russian), Vysshaya Shkola, Moscow, 1977.
[6] S. G. Michlin, Partielle Differentialgleichungen der mathematischen Physik, Akademie-Verlag, Berlin, 1978.
[7] D. Muzsi and R. Precup, Nonresonance theory for semilinear operator equations under regularity conditions, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 6(2008), 75–89.
[8] D. Muzsi and R. Precup, Non-resonance and existence for systems of non-linear operator equations, Appl. Anal. 87 (2008), 1005–1018.
[9] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Taylor and Francis, London, 2001.
[10] R. Precup, Existence results for non-linear boundary value problems under nonresonance conditions, in: Qualitative Problems for Differential Equations and Control Theory, C. Corduneanu (ed.), World Scientific, Singapore, 1995, pp. 263–273.
[11] R. Precup, Linear and Semilinear Partial Differential Equations, De Gruyter, Berlin, 2013.
[12] R. Precup, Nash-type equilibria and periodic solutions to nonvariational systems, Adv. Non-linear Anal. 3(2014), 197–207.
[13] B. Ricceri, Fixed points of nonexpansive potential operators in Hilbert spaces, Fixed Point Theory Appl. 2012 (2012), 123, 1–13.
[14] M. Struwe, Variational Methods, Springer, Berlin, 1990
2016

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