Existence and nonuniqueness of solutions of a singular nonlinear boundary-layer problem (Q810233)

The authors examine existence and uniqueness questions for positive solutions of \(gg''+h=0\) on (-k,1), \(k>0\), subject to the boundary conditions \(g'(-k)=C\) and \(g(1)=0\), where h is continuous, increasing, and (for some positive \(M_ 1\) and \(M_ 2)\) satisfies \(M_ 2| x| \leq h(x)\leq M_ 1| x|\). An important example arising in boundary layer theory is \(h(x)=x\). It is shown that for every real C, there are conditions on \(M_ 1\), \(M_ 2\), and k forcing nonuniqueness of solutions. Numerical results for \(h(x)=x\) are presented demonstrating this lack of uniqueness.