Uniformly boundedness of finite Morse index solutions to semilinear elliptic equations with rapidly growing nonlinearities in two dimensions

We consider the Gelfand problem with rapidly growing nonlinearities in two-dimensional bounded strictly convex domains. In this paper, we prove the uniform boundedness of finite Morse index solutions. As a result, we show that there exists a solution curve having infinitely many bifurcation points or turning points. These results are recently proved by Kumagai (2025) [1] for supercritical nonlinearities when the domain is the unit ball via an ODE argument. Instead of the ODE argument, we apply a new method focusing on the interaction between the growth condition of the nonlinearities and the shape of the fundamental solution of the Laplace equation. As a consequence, we clarify the bifurcation structure for general convex domains.