Semilinear Elliptic Equations on Star Graph with Small Edges

We consider a second order semilinear elliptic equation on a star graph with finitely many edges, some of which are supposed to have small lengths proportional to a small parameter $$\varepsilon.$$ At the common vertex, a general boundary condition is imposed, while at the boundary vertices, we impose the Dirichlet or Robin or Neumann condition. The nonlinearity in the equation depends only on the unknown function and variable and is supposed to satisfy a Lipschitz condition as well as a certain monotonicity condition. In addition, the nonlinearity on the small edges is multiplied by $$\varepsilon^{-1}.$$ The main results of the paper establish the unique solvability of the considered problem and describe the convergence and asymptotic expansions for the solution as $$\varepsilon$$ goes to zero. The limiting problem is considered on the graph without small edges and involves a certain limiting boundary condition at the common vertex. This limiting boundary condition depends in a nontrivial nonlinear way on the nonlinearities in the original equation on the small edges. The convergence is established uniformly in $$L_2$$ -norm of the right-hand side in the equation. We also construct the asymptotic expansion for the solution of perturbed problem up to an arbitrary power of $$\varepsilon,$$ and estimate the remainders in the asymptotics uniformly in $$L_2$$ -norm of the right-hand side in the equation.