Multiplicity of solutions for non-homogeneous Dirichlet problem with one-dimension Minkowski-curvature operator (Q2085999)

This paper explores the multiplicity of solutions for the following boundary value problem with one-dimensional Minkowski-curvature operator \[ \Biggl(\frac{u'}{\sqrt{1-\kappa u'^2}}\Biggr)'+f(u)=0,\;t\in(0,1), \] \[ u(0)=sA,\;\;u(1)=sB, \] where the constant \(\kappa\) is positive, \(A\) and \(B\) are constants, \(s\in\mathbb{R}\) is a parameter with \(|s(B-A)|<\frac{1}{\sqrt{\kappa}}\) and \(f\in C(\mathbb{R}, \mathbb{R})\) is such that \(f(s)s>0\) for \(s\ne0.\) In one of the main results, assuming in addition that there exists a positive integer \(j\) such that \(j^2\pi^2<f_0<(j+1)^2\pi^2,\) where \(f_0=\lim_{u\to 0}\frac{\sqrt{1-\kappa u^2}f(u)}{u},\) the authors establish that for every \(B>A>0\) there exist \(s_k\in(0,\frac{1}{\sqrt{\kappa}(B-A)}), k=0,1,...,2N-2,\) such that for every \(s\in(s _{2i-1}, s_{2i}), i=0,1,...,N-1,\) with \(s_{-1}=0,\) the considered problem has at least \(4(N-i)\) nontrivial solutions, where \(N\) is the number of positive integers \(m\) such that \(2m\in[2, j-\frac{3}{2}-\frac{\sqrt{2}}{2}\sqrt{\frac{A}{B-A}}\frac{1}{\pi}].\)