Abstract :
We consider the p-Laplacian boundary value problem where p>1 is a fixed number, p(s)=sp−2s, , and for each j=0,1, cj0+cj1>0. The function is a Carathéodory function satisfying, for , 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) and (2) is the problem together with (2), where a,b L1(0,1), , and u±(x)=max{±u(x),0} for x [0,1]. This problem is ‘positively-homogeneous’ and jumping. Values of λ for which (2) and (3) 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 (1) and (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) and (2).
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.
