Positive solutions to BVPs for a singular differential equation
The authors are concerned with existence of positive solutions for the singular second-order differential equation \[ \varphi''-\lambda\frac{| \varphi'| ^2}{\varphi}+f(t)=0,\quad t\in (0,1), \] subject to the boundary conditions \[ \varphi(1)=\varphi(0)=0\text{ or }\varphi'(1)=\varphi'(0)=0. \] In addition, the authors investigate uniqueness and asymptotic behavior of the solution as \(\lambda\to 0\) and as \(\lambda\to\infty\). The proof of the main results are based on the classical method of elliptic regularization. For related work, see [\textit{X. Zhang, L. Liu, Y. Wu}, Nonlinear Anal., Theory Methods Appl. 68, No.~1 (A), 97--108 (2008; Zbl 1135.34016)].
scientific article; zbMATH DE number 5152816