Remarks on semilinear problems with nonlinearities depending on the derivative

Abstract

In this paper, we continue some work by Cañada and Drábek [1] and Mawhin [6] on the range of the Neumann and Periodic boundary value problems: u''(t) + g(t, u'(t)) = f̄ + f̃(t), t ∈ (a, b) u'(a) = u'(b) = 0 or u(a) = u(b), u'(a) = u'(b) where g ∈ C ([a, b] x ℝn, ℝn), f̄ ∈ ℝn, and f̃ has mean value zero. For the Neumann problem with n > 1, we prove that for a fixed f̃ the range can contain an infinity continuum. For the one dimensional case, we study the asymptotic behavior of the range in both problems.