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| We study the exact multiplicity and bifurcation diagrams of positive solutions $u\in C^{2}(-L, L)\cap C[-L, L]$ of the one-dimensional multiparameter prescribed mean curvature problem $-\left( \frac{u'(x)}{ \sqrt{ 1+ \left( u'(x) \right)^2 } } \right)'= \lambda (u^p+ u^q)$, $x\in (-L, L)$, $u(-L)= u(L)=0,$ where $\lambda >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, and $0\leq p< q< \infty$ are two constants. We prove that the problem has at most two positive solutions for any $0\leq p< q< \infty$ and $\lambda , L>0$. In addition, if $0\leq p< q \leq \tilde{q}(p)=p+1+2\sqrt{p+1}$, we give a classification of totally three qualitatively different bifurcation diagrams on the $(\lambda ,\left\| u \right\|_{\infty} )$-plane for any $L>0$. For any fixed $p\geq 0$ and $q\geq \bar{q}(p)=p+2+2\sqrt{2p+3}$, we prove that there exist positive $L_*< L^*$ such that the bifurcation diagrams on the $( \lambda ,\left\| u \right\|_{\infty } )$-plane are individually qualitatively different for the cases (i) $L\in (0, L_*)$ and $L> L^*$, (ii) $L=L^*$, (iii) $ L \in [ L_*, L^*)$. |
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