Sharp parameter ranges in the uniform anti-maximum principle for second-order ordinary differential operators
The author studies the differential equation \((pu')'- qu+\lambda wu= f\) in \((0,1)\) with the boundary conditions \((\alpha u+ pu')|_{x=0}= 0\), \((\beta u+ pu')|_{x=1}= 0\), where constants \(\alpha,\beta\in\mathbb{R}\), \(p,q,w\in C([0,1],\mathbb{R})\) and \(p\), \(w> 0\) in \([0,1]\). Let \(\lambda^{\alpha\beta}\) be the first eigenvalue of the corresponding Sturm-Liouville problem and let \(\lambda^{\infty\beta}\), \(\lambda^{\alpha,\infty}\) stand for the first eigenvalue with zero Dirichlet boundary condition at \(x= 0\), \(x= 1\), respectively, and unchanged boundary condition at the opposite endpoint. He shows that if the parameter \(\lambda\) in the above equation belongs to the interval \((\lambda^{\alpha,\beta}_1,\,\min\{\lambda^{\infty\beta}, \lambda^{\alpha\infty}\}]\) and if the function \(f\) is \(\leq 0\) and not identically zero, then the solution \(u\) of the above problem is negative in \([0,1]\).
scientific article; zbMATH DE number 2021116