Existence of solutions of semilinear systems in ℓ²
Keywords:Graphs, partial difference equations, nonlinear elliptic equations, Laplacian, grafos, ecuaciones diferenciales parciales, ecuaciones elípticas no-lineales, laplacianas.
Let Q : ℓ² → ℓ² be a symmetric and positive semi-definite linear operator and fj : R → R (j = 1, 2, ...) be real functions so that, fj(0) = 0 and, for every x = (x1, x2, ....) ∈ ℓ², it holds that f(x) := (f1(x1), f2(x2), ...) ∈ ℓ². 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 ℓ² 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.
 Bapat, R. B. The Laplacian Matrix of a Graph. Math. Student 65, pp. 214-223, (1996).
 Colin de Verdiere, Y. Spectre d’operateurs différentiels sur les graphes. In Random walks and discrete potential theory, Cortona, June 22-28, pp. 1-26, (1997).
 Colin de Verdiere, Y. Spectres de graphes. Societé Mathématique de France (1998).
 Friedrichs, K. and Lewy, H. Uber die partiellen Differenzengleichungen der mathematischen Physik. (German) Math. Ann. 100 (1), pp. 32-74,
 Hidalgo, R. A. Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs. To appear in Journal of Difference Equations and Applications.
 Howe, M. S. Acoustics of fluid-structure interactions. Cambridge, New York. Cambridge University Press (1998).
 Matukuma, T. The Cosmos. Iwanami Shoten, Tokio, (1938).
 Mohar, B. The laplacian spectrum of graphs. In Graph Theory, Combinatorics, and Applications 2. Ed. Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk. Wiley, pp. 871-898, (1991).
 Neuberger, John M. Nonlinear Elliptic Partial Difference Equations on Graphs. Experimental Mathematics 15, pp. 91-107, (2006).
 Yi, Li. On the positive solutions of the Matukuma equation. Duke Math. J. 70 (3), pp. 575-589, (1993).