Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation.

*(English)*Zbl 1263.34028The paper is concerned with the existence and multiplicity of solutions of the quasi-linear equation
\[
-(u'/\sqrt{1-u'{}^{2}})'=f(t,u),\quad 0<t<T,
\]
subject to the Dirichlet boundary conditions \(u(0)=u(T)=0\). The authors use transformations to rewrite the above equation either as
\[
-u''=g(t,u)h(u'),
\]
where \(g\) is bounded and \(h\) has compact support, or as
\[
-(\psi(u'))'=g(t,u),
\]
where \(\psi\) is an asymptotically linear homeomorphism on the real line, and \(g\) is bounded. Then depending on the behaviour of the nonlinearity \(f=f(t,s)\) near \(s=0\) (\(f\) is not necessarily positive), the authors prove the existence of either one, or two, or three, or infinitely many positive solutions. For this, they employ topological and variational methods.

Reviewer: Smail Djebali (Algiers)

##### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34C23 | Bifurcation theory for ordinary differential equations |

47J30 | Variational methods involving nonlinear operators |

47N20 | Applications of operator theory to differential and integral equations |

##### Keywords:

quasilinear ordinary differential equation; Minkowski-curvature; Dirichlet boundary condition; positive solution; existence; multiplicity; critical point theory; bifurcation method
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\textit{I. Coelho} et al., Adv. Nonlinear Stud. 12, No. 3, 621--638 (2012; Zbl 1263.34028)

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