Semilinear Elliptic Equations on Thin Network-Shaped Domains with Variable Thickness

We study semilinear elliptic equations in a thin domain which is shaped like a network and degenerate into a geometric graph when a certain parameter tends to zero. It is shown by using a comparison technique that a solution of the equation in the network-shaped domain approaches a solution of an associated limit equation on the graph. Conversely, when the limit equation on the graph has a solution, we prove the existence of a solution of the PDE in the network-shaped domain which converges to the solution of the limit equation.