Ambrosetti-Prodi type results for a Neumann problem with a mean curvature operator in Minkowski spaces (Q2211933)

This paper discusses the solvability of the following Neumann boundary value problem with mean curvature operator in the Minkowski space: \[\Bigl(\frac{u'}{\sqrt{1-u'^2}}\Bigr)'+a(x)g(u)=\mu+p(x),\;x\in(0,T),\] \[u'(0)=0=u'(T),\] where \(a\in L^{\infty}(0,T),\) \(a(x)\geq0\) on \((0,T)\) and \(\int_0^Ta(x)dx>0,\) \(g\in C^1(\mathbb{R},\mathbb{R})\) is such that \(g(0)=0,\) \(g'(s)<0\) for any \(s<0\) and \[-\infty<\lim_{s\to-\infty}g'(s)<0<\lim_{s\to+\infty}g'(s)<+\infty,\] \(\mu\in\mathbb{R},\) and \(p\in L^{\infty}(0,T).\) Using the shooting method, the authors show that there is \(\mu^*\in\mathbb{R}\) such that the considered problem has at least two solutions for all \(\mu>\mu^*.\)