p-Laplacian problems with jumping nonlinearities

Abstract

We consider the p-Laplacian boundary value problemc 00 u(0) = c 01 u (0), c 10 u(1) = c 11 u (1),where p > 1 is a fixed number, φ 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] × R 2 → R is a Carathéodory function satisfying, for (x, s, t) ∈ [0, 1] × R 2 ,where ψ ± , Ψ ± ∈ L 1 (0, 1), and E has the form E(x, s, t) = ζ(x)e(|s| + |t|), with ζ ∈ L 1 (0, 1), ζ 0, e 0 and lim r→∞ e(r)r 1−p = 0. This condition allows the nonlinearity in (1) to behave differently as u → ±∞. Such a nonlinearity is often termed jumping.together with (2), where a, b ∈ L 1 (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 nontrivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called halfeigenfunctions.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 '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.

Nonresonance conditions for generalised ϕ-Laplacian problems with jumping nonlinearities

“…The proofs involving the other integral equations are similar. 1], and in this case the similar (but somewhat simpler) integral equation formulation used in [15] can also be used here.…”

“…In the case of the p-Laplacian (and more particularly, the semilinear case p = 2), such conditions have been discussed extensively in the literature, with nonresonance conditions described in terms of either eigenvalues of the p-Laplacian (for non-jumping f ) or the Fučík spectrum or halfeigenvalues of the p-Laplacian (for jumping f ). A survey of such results is given in Section 7 of [15] (and in Section 1 of [3] for the case p = 2), so we omit a detailed discussion of this case here. However, to introduce some terminology and to motivate our discussion of the general ψ -Laplacian problem, we will give a brief description of the p-Laplacian results in the following paragraph.…”

“…Values of λ for which (1.2), (1.5), have a non-trivial solution u will be called half-eigenvalues of the p-Laplacian, and the solutions u are half-eigenfunctions (if m + = m − these are eigenvalues and eigenfunctions respectively). Nonresonance conditions for jumping f , with general L 1 coefficients ξ ± , Ξ ± , are obtained in [15] in terms of half-eigenvalues of the p-Laplacian; other types of nonresonance conditions are described in [15,Section 7], and in the references therein. Nonresonance conditions for non-jumping f , in terms of eigenvalues, are described in, for example, [1,10].…”

“…Nonresonance conditions for non-jumping f , in terms of eigenvalues, are described in, for example, [1,10]. When f is jumping, but the functions ξ ± , Ξ ± are constants, nonresonance conditions have been given in terms of the so-called 'Fučík spectrum', see the references in [15]. However, it is shown in [15] that in this case the half-eigenvalue and Fučík spectrum conditions are equivalent.…”

Multiplicity results and global bifurcations for a degenerate elliptic equation

“…Even there has been a significant amount of work that has been done on degenerate elliptic equations, and in particular, bifurcation problems for pLaplacian equations, not only for one-dimensional problems but in higher dimensions as well, in the last 25 years. Authors who have contributed to this field include, for instance, Laurent Veron, M.Guedda [16], Pavel Drabek [3], [8], [9], [10], [11], [12], Bryan Rynne [4], [5], [13], [14], [15], and many others. However, until recently, the basic spectral properties obtained in the case p > 2 and the exact numbers of solutions of problem E p λ and especially the existence of collection of intervals I n , n = 1, 2, 3 .…”