Abstract
A Lyapunov-type inequality is established for the partial differential equation:
-G(gamma) u(x, y) = w(x)u(x, y), (x, y) is an element of Omega,
u(x, y) = 0, (x, y) is an element of partial derivative Omega,
where Omega = [a, b] x O, (a, b) is an element of R-2, a < b, O, is an open bounded subset in R-N, N >= 1, G(gamma) is the differential operator given by G(gamma) u(x, y) := (partial derivative(2)u/partial derivative x(2)) + x(2 gamma) Delta(gamma)u, gamma >= 0, (x, y) is an element of Omega, and w is an element of C([a, b]). Next, some sign-changing criteria are obtained for the nontrivial solutions to the considered problem.