Mathew Gluck
Bifurcation results for nonlinear problems involving the graph Laplacian
Abstract: This note illustrates some standard bifurcation results in nonlinear functional analysis in a setting that welcomes those with no experience in functional analysis. Specifically, I will show that for a suitable family of nonlinear PDEs involving the graph Laplacian on a finite connected simple graph, the first eigenvalue of the graph Laplacian is a bifurcation point. Moreover, there is a smooth parameterized curve of nontrivial solutions bifurcating from the trivial solution at the first eigenvalue.
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