p-Laplacian problems with jumping nonlinearities

We consider the p-Laplacian boundary value problem{A formula is presented}{A formula is presented} where p > 1 is a fixed number, f{symbol}_p ( s ) = | s |^{p - 2} s, s ? R, and for each j = 0, 1, | c_{j 0} | + | c_{j 1} | > 0. The function f : [ 0, 1 ] × R² ? R is a Carathéodory function satisfying, for ( x, s, t ) ? [ 0, 1 ] × R²,{A formula is presented} where ?_±, ?_± ? L¹ ( 0, 1 ), and E has the form E ( x, s, t ) = ? ( x ) e ( | s | + | t | ), with ? ? L¹ ( 0, 1 ), ? {greater than or slanted equal to} 0, e {greater than or slanted equal to} 0 and lim_{r ? 8} e ( r ) r^{1 - p} = 0. This condition allows the nonlinearity in (1) to behave differently as u ? ± 8. Such a nonlinearity is often termed jumping. Related to (1), (2) is the problem{A formula is presented} together with (2), where a, b ? L¹ ( 0, 1 ), ? ? R, and u^± ( x ) = max { ± u ( x ), 0 } for x ? [ 0, 1 ]. This problem is 'positively-homogeneous' and jumping. Values of ? for which (2), (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), (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 'Fucík 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. © 2005 Elsevier Inc. All rights reserved.