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>Half-eigenvalues of self-adjoint, 2m th-order differential operators and semilinear problems with jumping nonlinearities
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Half-eigenvalues of self-adjoint, 2m th-order differential operators and semilinear problems with jumping nonlinearities
We consider semilinear boundary value problems of the form Lu(x) = f(x,u(x)) + h(x), x ∈ (0, π), (1) where L is a 2mth-order, self-adjoint, disconjugate ordinary differential operator on [0, π], together with appropriate boundary conditions at 0 and π, while f:[0, π] * R → R is a Caratheodory function and h ∈ L~2(0, π). We assume that the limits a(x) := lim_(ξ→∞) f(x,ξ)/ξ, b(x) := lim_(ξ→-∞ )f(x,ξ)/ξ, exist for almost every x ∈ [0, π] and a, b ∈ L~∞ (0,π), but a ≠ b. In this case the nonlinearity f is termed jumping. Closely related to (1) is the "limiting" boundary value problem Lu = au~+ - bu~- + λu + h, (2) where u~±(x) = max{±u(x), 0} for x ∈[0, π], and λ is a real parameter. Values of λ for which (2) (with h = 0) has a nontrivial solution u will be called half-eigenvalues of (L; a, b). In this paper we show that a sequence of half-eigenvalues exists, with certain properties, and we prove various results regarding the existence and multiplicity of solutions of both (1) and (2). These result depend strongly on the location of the half-eigenvalues relative to the point λ = 0. Some geometric properties of the Fucik spectrum of L are also briefly discussed.
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