Uniqueness of positive solutions for singular nonlinear boundary value problems
The authors consider the uniqueness of positive solutions of the singular nonlinear differential equation \[ u''(r)+{\textstyle \frac mr} u'(r)+f(r,u(r))=0 \quad\text{in }(t_1,t_2), \qquad m\geq 0,\;0\leq t_1<t_2<\infty \] with one of the following sets of boundary conditions \[ u(t_1)= k_1\geq 0,\quad u(t_2)= k_2\geq 0, \qquad u'(t_1)= u(t_2)=0, \] \[ a_1u(t_1)+ b_1u'(t_1)=0, \qquad a_2u(t_2)+ b_2u'(t_2)=0, \] where \(a_i,b_i\in (-\infty,\infty)\) satisfy \(a_i^2+b_i^2\neq 0\), \(i=1,2\), \(u\in C^1[t_1,t_2]\), \(f:(t_1,t_2)\times (0,\infty)\to(- \infty,\infty)\) is local Lipschitz continuous in \(u\in(0,\infty)\) for each fixed \(r\in(t_1,t_2)\), \(f(r,u)/u\) is strictly decreasing with respect to \(u\in(0,\infty)\) for each fixed \(r\in(t_1,t_2)\). The purpose of the paper is to give an extension of known results.
singular nonlinear differential equation
scientific article; zbMATH DE number 986951