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Exact number of positive solutions of the Dirichlet problem for the one-dimensional prescribed mean curvature equations. (English) Zbl 1348.34054

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
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