p-Laplacian problems with jumping nonlinearities

We consider the p-Laplacian boundary value problemequation(1)−(ϕp(u′(x)))′=f(x,u(x),u′(x)),a.e.x∈(0,1),equation(2)c00u(0)=c01u′(0),c10u(1)=c11u′(1), where p>1p>1 is a fixed number, ϕp(s)=|s|p−2sϕp(s)=|s|p−2s, s∈Rs∈R, and for each j=0,1j=0,1, |cj0|+|cj1|>0|cj0|+|cj1|>0. The function f:[0,1]×R2→R is a Carathéodory function satisfying, for (x,s,t)∈[0,1]×R2(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±, Ψ±∈L1(0,1)Ψ±∈L1(0,1), and E   has the form E(x,s,t)=ζ(x)e(|s|+|t|)E(x,s,t)=ζ(x)e(|s|+|t|), with ζ∈L1(0,1)ζ∈L1(0,1), ζ⩾0ζ⩾0, e⩾0e⩾0 and limr→∞e(r)r1−p=0limr→∞e(r)r1−p=0. This condition allows the nonlinearity in (1) to behave differently as u→±∞u→±∞. Such a nonlinearity is often termed jumping.Related to  and  is the problemequation(3)−(ϕp(u′)′)=aϕp(u+)−bϕp(u−)+λϕp(u),in (0,1), together with (2), where a,b∈L1(0,1)a,b∈L1(0,1), λ∈Rλ∈R, and u±(x)=max{±u(x),0}u±(x)=max{±u(x),0} for x∈[0,1]x∈[0,1]. This problem is ‘positively-homogeneous’ and jumping. Values of λ for which  and  has a nontrivial 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 half-eigenfunctions 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 nonexistence results for the problem  and . 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  and .When the functions a and b are constant the set of half-eigenvalues is closely related to the ‘Fučík 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.