Cheng, Hongmei; Yan, Baoqiang Exact number of positive solutions of the Dirichlet problem for the one-dimensional prescribed mean curvature equations. (English) Zbl 1348.34054 Commun. Appl. Anal. 18, No. 1-2, 345-364 (2014). Summary: This paper presents the exact number of positive solutions of the Dirichlet problem for the one-dimensional prescribed mean curvature equations \[ -\Biggl({u'\over \sqrt{1+(u')^2}}\Biggr)'= \lambda(u^p+ u^q+ ku),\quad u(0)= u(1)= 0, \] where \(0<p<q<+\infty\) which satisfy either \(0<p<q<1\) or \(1<p<q<+\infty\) and \(k>0\) are fixed given numbers, and \(\lambda>0\) is a parameter. And we obtain that when \(0<p<q<1\), \(k>0\), \(\lambda>0\), the problem has at most two solutions and there exist \(9<\lambda_1<\lambda_2<\infty\) such that the problem has one solution \(0<\lambda<\lambda_1\) and has no solution for \(\lambda>\lambda_2\); when \(1<p<q<+\infty\), \(k>0\), \(\lambda>0\), and \(0<\lambda k<\pi^2\) the problem has at most one s olution and there exist \(0<\lambda_1<\lambda_2<+\infty\) such that the problem has no solution for \(0<\lambda<\lambda_1\) and has one solution for \(\lambda>\lambda_2\). The arguments are based upon a time-map method. MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B09 Boundary eigenvalue problems for ordinary differential equations Keywords:Dirichlet problem; mean curvature equations; time-map method PDFBibTeX XMLCite \textit{H. Cheng} and \textit{B. Yan}, Commun. Appl. Anal. 18, No. 1--2, 345--364 (2014; Zbl 1348.34054)