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

TL;DR

This paper proves that solutions with finite Morse index to certain semilinear elliptic equations in two dimensions are uniformly bounded, revealing complex bifurcation structures in convex domains.

Contribution

It introduces a new method to establish boundedness of solutions, extending previous results from the unit ball to general convex domains.

Findings

Finite Morse index solutions are uniformly bounded in 2D.

Existence of solution curves with infinitely many bifurcation points.

Clarification of bifurcation structures in convex domains.

Abstract

We consider the Gelfand problem with rapidly growing nonlinearities in the two-dimensional bounded strictly convex domains. In this paper, we prove the uniformly boundedness of finite Morse index solutions. As a result, we show that there exists a solution curve having infinitely many bifurcation/turning points. These results are recently proved by the present author 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. As a result, we clarify the bifurcation structure for general convex domains.