Sharp parameter ranges in the uniform anti-maximum principle forsecond-order ordinary differential operators

We consider the equation(pu)-qu+wu = fin (0,1) subject to homogenous boundary conditions atx = 0andx = 1, e.g.,u(0) = u(1) = 0. Let₁be the first eigenvalue of the corresponding Sturm-Liouville problem. Iff 0but 0 then it is known that there exists > 0 (independent onf) such that for (₁, ₁ + ]any solutionumust be negative. This so-calleduniform anti-maximum principle(UAMP)goes back to Clément, Peletier [4]. In this paper weestablish the sharp values of for which (UAMP) holds. The same phenomenon, including sharp values of, can be shown for the radially symmetricp-Laplacian on balls and annuli inⁿprovided1 n < p. The results are illustrated by explicitly computed examples.