Nonresonance between the first two eigenvalues of a quasi-linear problem

We consider the quasilinear Dirichlet boundary value problem -(phi(p)(u'))' = f(u) + h(x), u(a) = u(b) = 0, where (phi(p)(u'))' is the one dimentional p-Laplacien, and prove its solvability for any given h, under new assumptions on the asymptotic behaviour of the potential of the nonlinearity f, with respect to two first eigenvalues of the associated quasilinear problem.