Lyapunov-type inequalities for one-dimensional Minkowski-curvature problems
In this paper, some Lyapunov-type inequalities for the one-dimensional Minkowski-curvature problem of the form \[ \begin{aligned} -&(\phi(u'(t)))'=r(t)u(t),\qquad a<t<b\\ &u(a)=u(b)=0, \end{aligned} \] where \[ \phi(z):=\frac{z}{\sqrt{1-|z|^2}},\qquad z\in(-1,1), \] the weight function \(r(t)\) (\(\geq0\) for all \(t\in(a,b)\)) satisfies that \(r\not\equiv0\) in any compact subinterval of \([a,b]\) and \[ r\in{\mathcal A}:=\{r\in L^1_{\mathrm{loc}}((a,b),[0,\infty)):\int^b_a(s-a)(b-s)r(s)\mathrm{d}s<\infty\}. \] The class \({\mathcal A}\) admits rather stronger singular functions at the boundary. As far as it is known, Lyapunov-type inequalities of Minkowski-curvature problems have not been studied. In the current paper, Lyapunov-type inequalities are established to give necessary conditions for the existence of positive solutions for scalar equations as well as systems of one-dimensional Minkowski-curvature problems with singular weights which belong to \({\mathcal A}\). In the paper, Lyapunov inequalities for one-dimensional Minkowski-curvature problem with a singular weight function, for one-dimensional Minkowski-curvature problem which possesses a singular coefficient and a singular weight which is transformed from problems to study radial solutions in exterior domain of a ball, and for a cycled and a strongly coupled system of one-dimensional Minkowski-curvature problem with singular weights are also derived.
scientific article; zbMATH DE number 7026153