The existence of positive solutions to a singular nonlinear boundary value problem
The paper deals with the problem \(-u''(t)= k(t) [u(t)]^{-\alpha} [u'(t)]^ \sigma\), \(u(0)=0\), \(u(1)=\Phi(u'(1))\). Making use of the shooting method the authors show that the given problem possesses a positive solution if the following assumptions are satisfied: \(\alpha>0\), \(\sigma<\alpha+1\), the function \(k(t)\) is nonnegative and measurable on \([0,1]\) and such that \(0<\int_ 0^ 1 k(t) dt<\infty\), the function \(\Phi(\beta)\) is positive and continuous on \((0,\infty)\) and such that \(\lim_{\beta\to \infty}\Phi(\beta)\beta^{-1}=D>1\), \(\lim_{\beta\to 0+} [\Phi(\beta)]^ \alpha \beta^{1-\sigma}=0\) if \(\sigma>1\), \(\lim_{\beta\to 0+} [\Phi(\beta]^ \alpha \ln(\beta)=0\) if \(\sigma=1\) and \(\lim_{\beta\to 0+} \Phi(\beta)=0\) if \(\sigma<1\).
singular nonlinear boundary value problem
scientific article; zbMATH DE number 399070