  We consider the Gelfand problem with general supercritical nonlinearities in
the two-dimensional unit ball. In this paper, we prove the non-existence of an
unstable solution for any small parameter and the uniformly boundedness of
finite Morse index solutions. As a result, we obtain the existence of a radial
singular solution and prove that the bifurcation curve has infinitely many
turning points. We remark that these properties are well-known in $N$
dimensions with $3\le N \le 9$ and less known in two dimensions. The key idea
of the proof is to utilize an interaction between a key gradient estimate of
solutions and the supercriticality of the nonlinearities.
