This paper investigates the bifurcation curves and the multiplicity of positive solutions for a one-dimensional Minkowski curvature problem with constant-yield harvesting
$ \begin{equation*} \left\{ \begin{array}{l} -\left( u^{\prime }(x)/\sqrt{1-\left[ {u^{\prime }}(x)\right] ^{2}}\right) ^{\prime } = \lambda g(u(x))-\mu ,\text{ for }x\in \left( -L,L\right) , \\ u(-L) = u(L) = 0, \end{array} \right. \end{equation*} $
where $ \lambda, L, \mu > 0 $ and there exist constants $ \sigma > u_{0} > 0 $ such that $ g\in C[0, \sigma ]\cap C^{2}(0, \sigma) $, $ g(0) = g(\sigma) = 0 $, $ g^{\prime }(u) > 0 $ on $ \left(0, u_{0}\right) $, $ g^{\prime }(u_{0}) = 0 $, $ $ and $ g^{\prime }(u) < 0 $ on $ (u_{0}, \sigma) $. We first show that the bifurcation curve has a $ \subset $-like shape and then provide additional sufficient conditions under which the curve is exactly $ \subset $-shaped. These results yield the exact multiplicity of positive solutions. As an application to population and ecological models, we further consider the nonlinearity $ g(u) = u^{p}\left(1-u^{q}/K\right) ^{r} $, where $ p, q, r, K > 0. $
Citation: Shao-Yuan Huang. Multiplicity of positive solutions for Minkowski curvature problems arising from population models with constant-yield harvesting[J]. Electronic Research Archive, 2026, 34(7): 5062-5086. doi: 10.3934/era.2026224
This paper investigates the bifurcation curves and the multiplicity of positive solutions for a one-dimensional Minkowski curvature problem with constant-yield harvesting
$ \begin{equation*} \left\{ \begin{array}{l} -\left( u^{\prime }(x)/\sqrt{1-\left[ {u^{\prime }}(x)\right] ^{2}}\right) ^{\prime } = \lambda g(u(x))-\mu ,\text{ for }x\in \left( -L,L\right) , \\ u(-L) = u(L) = 0, \end{array} \right. \end{equation*} $
where $ \lambda, L, \mu > 0 $ and there exist constants $ \sigma > u_{0} > 0 $ such that $ g\in C[0, \sigma ]\cap C^{2}(0, \sigma) $, $ g(0) = g(\sigma) = 0 $, $ g^{\prime }(u) > 0 $ on $ \left(0, u_{0}\right) $, $ g^{\prime }(u_{0}) = 0 $, $ $ and $ g^{\prime }(u) < 0 $ on $ (u_{0}, \sigma) $. We first show that the bifurcation curve has a $ \subset $-like shape and then provide additional sufficient conditions under which the curve is exactly $ \subset $-shaped. These results yield the exact multiplicity of positive solutions. As an application to population and ecological models, we further consider the nonlinearity $ g(u) = u^{p}\left(1-u^{q}/K\right) ^{r} $, where $ p, q, r, K > 0. $
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Shao-Yuan Huang. Multiplicity of positive solutions for Minkowski curvature problems arising from population models with constant-yield harvesting[J]. Electronic Research Archive, 2026, 34(7): 5062-5086. doi: 10.3934/era.2026224
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