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Evolution of bifurcation curves for one-dimensional Minkowski-curvature problem. (English) Zbl 1441.53010

Summary: This paper is concerned with the Robin problem for the prescribed mean curvature equation in Minkowski space \[\begin{cases} - \bigg(u^\prime / \sqrt{ 1 - | u^\prime |^2}\bigg)^\prime = \lambda u^q + \mu u^p, \quad t \in (0, L), \\ u^\prime (0) = u (L) = 0, \end{cases} \tag{P}\] where \(0 < q < 1 < p\). We show that there exists a constant \(\mu^\ast > 0\) and two functions \(\Lambda_\ast (\cdot), \Lambda^\ast (\cdot)\) with \[\Lambda_\ast (\mu) < 0 < \Lambda^\ast (\mu), \quad\mu > \mu^\ast,\] such that for every \(\mu > \mu^\ast\) and for all \(\lambda \in (\Lambda_\ast (\mu), 0)\), (P) has at least two positive solutions; for every \(\mu > \mu^\ast\) and for all \(\lambda \in (0, \Lambda^\ast (\mu))\), (P) has at least three positive solutions. The proof combines topological degree and bifurcation technique. We also present a numerical computation of the bifurcation curves.

MSC:

53A35 Non-Euclidean differential geometry
53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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