Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions

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 positive small parameter $\lambda$. The result implies that once the bifurcation curve emanates from the starting point, then the curve never approaches $\lambda =0$. As a result, we obtain the existence of a radial singular solution. In addition, we prove the uniformly boundedness of finite Morse index solutions. As a result, we 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. Our results clarify that the bifurcation structure is solely determined by the supercriticality of the nonlinearities if $2\le N\le 9$.