An uncountable number of solutions for a BVP governing Marangoni convection

This paper investigates a boundary value problem (BVP) governing Marangoni convection over a flat surface. The BVP involves a temperature gradient parameter k>-1. Previous numerical and analytical studies report the existence of one solution for each value of k. Here we show that the nature of the solutions varies greatly depending on the value of k. For each -1<k<-1/2 we prove that an uncountable number of solutions to the BVP exist. For -1/2≤k<0 a unique solution exists. For k≥0 we prove the existence of a solution but, based on numerical integration of the differential equation, conjecture the existence of precisely two solutions when k>0.