## 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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