Exact Multiplicity and S-Shaped Bifurcation Curve for some Semilinear Elliptic Problems from Combustion Theory

Abstract

In this paper, by making use of a new limiting equation and a continuation method based on a local bifurcation theorem of Crandall and Rabinowitz, we rigorously confirm some long-standing conjectures on the exact number of positive solutions for a class of elliptic equations arising from combustion theory. This work extends that of [S.-H. Wang, Proc. Roy. Soc. London Sect. A, 454 (1998), pp. 1031--1048] for the 1 dimension case to cover both dimensions 1 and 2, and it extends the work of [Y. Du and Y. Lou, J. Differential Equations, to appear] for the special case m=0 to $0\leq m < 1$. It is shown that the main results are not true if the dimension is greater than 2 or if $m\geq 1$. Therefore, our results are in a sense the best possible.