## versión impresa ISSN 0716-0917

### Proyecciones (Antofagasta) v.27 n.2 Antofagasta ago. 2008

#### http://dx.doi.org/10.4067/S0716-09172008000200004

Proyecciones Journal of Mathematics
Vol. 27, Nº 2, pp. 171-183, August 2008.
Antofagasta - Chile

EXISTENCE OF SOLUTIONS OF SEMILINEAR SYSTEMS IN

RUBÉN HIDALGO 1
MAURICIO GODOY 2

1 Universidad Técnica Federico Santa María, Chile.
2 University of Bergen, Norway.

Correspondencia a:

Abstract
Let Q : be a symmetric and positive semi-definite linear operator and fj : (j = 1, 2, ...) be real functions so that, fj(0) = 0 and, for every x = (x1, x2, ....) , it holds that f (x) := (f1(x1), f2(x2), ...) . Sufficient conditions for the existence of non-trivial solutions to the semilinear problem Qx = f (x) are provided. Moreover, if G is a group of orthogonal linear automorphisms of which commute with Q, then such sufficient conditions ensure the existence of non-trivial solutions which are invariant under G. As a consequence, sufficient conditions to ensure solutions of nonlinear partial difference equations on finite degree graphs with vertex set being either finite or infinitely countable are obtained. We consider adaptations to graphs of both Matukuma type equations and Helmholtz equations and study the existence of their solutions.

Key words : Graphs, Partial difference equations, Nonlinear elliptic equations, Laplacian.
Subjclass :
[2000] 05C12, 39A12, 35J05.

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RUBÉN HIDALGO
Departamento de Matemáticas
Valparaíso
Chile
e-mail : ruben.hidalgo@usm.cl

MAURICIO GODOY
Department of Mathematics
University of Bergen
Bergen
Norway
e-mail : mauricio.godoy@gmail.com

Received : March 2008. Accepted : July 2008

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