Sufficient conditions for exact bifurcation curves in Minkowski curvature problems and their applications

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Abstract

In this paper, we continued and extended the work in [1–3] by establishing sufficient conditions to determine the exact shape of the bifurcation curve of positive solutions for the Minkowski curvature problem

$\left\{\begin{array}{l} -\left(u^{\prime} / \sqrt{1-u^{\prime 2}}\right)^{\prime}=\lambda f(u), \text { in }(-L, L), \\ u(-L)=u(L)=0, \end{array}\right.$

where $\lambda, L>0 \text { and } f \in C^{2}(0, \infty)$. Notice that we allow $f\left(0^{+}\right)=-\infty$. To illustrate the applicability of these results, we presented some examples, such as the diffusive logistic equation with a Holling type-II functional response.

Keywords

positive solution / bifurcation curve / Minkowski curvature problem / diffusive logistic equation

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