p-LAPLACIAN PROBLEMS WITH JUMPING NONLINEARITIES

We consider the p-Laplacian boundary value problem −(φp(u(x)) = f(x, u(x), u′(x)), a.e. x ∈ (0, 1), (1) c00u(0) + c01u ′(0) = 0, c10u(1) + c11u ′(1) = 0, (2) where p > 1 is a fixed number, φp(s) = |s|p−2s, s ∈ R, and for each j = 0, 1, |cj0|+ |cj1| > 0. The function f : [0, 1]× R2 → R is a Carathéodory function satisfying, for (x, s, t) ∈ [0, 1]× R2, ψ±(x)φp(s)− E(x, s, t) ≤ f(x, s, t) ≤ Ψ±(x)φp(s) + E(x, s, t), ±s ≥ 0, where ψ±, Ψ± ∈ L1(0, 1), and E has the form E(x, s, t) = ζ(x)e(|s|+ |t|), with ζ ∈ L1(0, 1), ζ ≥ 0, e ≥ 0 and limr→∞ e(r)r1−p = 0. This condition allows the nonlinearity in (1) to behave differently as u→ ±∞. Such a nonlinearity is often termed jumping. Related to (1), (2) is the problem −(φp(u) = aφp(u)− bφp(u) + λφp(u), in (0, 1), (3) together with (2), where a, b ∈ L1(0, 1), λ ∈ R, and u±(x) = max{±u(x), 0} for x ∈ [0, 1]. This problem is ‘positively-homogeneous’ and jumping. Values of λ for which (2), (3) has a non-trivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding halfeigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to existence and non-existence results for the problem (1), (2). We also consider a related bifurcation problem, and obtain a global bifurcation result similar to the well-known Rabinowitz global bifurcation theorem. This then leads to a multiplicity result for (1), (2). When the functions a and b are constant the set of half-eigenvalues is closely related to the ‘Fuč́ık spectrum’ of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results.