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Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation. (English) Zbl 1263.34028
The 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.

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
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