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Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities. (English) Zbl 1250.34020
Summary: The exact number of solutions is obtained for the one-dimensional prescribed mean curvature equation with concave-convex nonlinearities in the form of \[ \begin{cases} -(\frac {u'}{\sqrt{1+{u'}^2}})' = \lambda (u^p +u^q), \\ u(x)>0, 0<x<1, \\ u(0) = u(1) =0, \end{cases} \] where \(\lambda >0\) is a parameter and \(p,q\) satisfy \(0<p<1<q<+\infty\). The arguments are based upon a time-map method.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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