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Nonlinear elliptic partial difference equations on graphs. (English) Zbl 1160.35412

Summary: This article furthers the study of nonlinear elliptic partial difference equations (PdE) on graphs. We seek solutions \(u : V \to\mathbb R\) to the semilinear elliptic partial difference equation \(-Lu+f(u) = 0\) on a graph \(G = (V,E)\), where \(L\) is the (negative) Laplacian on the graph \(G\). We extend techniques used to prove existence theorems and derive numerical algorithms for the partial differential equation (PDE) \(\Delta u+f(u) = 0\). In particular, we prove the existence of sign-changing solutions and solutions with symmetry in the superlinear case. Developing variants of the mountain pass, modified mountain pass, and gradient Newton-Galerkin algorithms for our discrete nonlinear problem, we compute and describe many solutions. Letting \(f = f(\lambda, u)\), we construct bifurcation diagrams and relate the results to the developed theory.

MSC:

35J61 Semilinear elliptic equations
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
39A14 Partial difference equations
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