ON A PSEUDO FUCIK SPECTRUM FOR STRONGLY NONLINEAR 2ND-ORDER ODES AND AN EXISTENCE RESULT

[摘要] We study here the existence of solutions to the nonlinear Dirichlet problem (P): (phi(u'))' + f(t, u) = q(t), a.e. for t is an element of [a, b], u(a)=u(b)=0, where phi is an increasing odd homeomorphism of R, f:[a, b]XR --> R satisfies the Caratheodory assumptions and q is an element of L(1)([a, b], R). With a view towards defining a pseudo Fucik spectrum (PFS) we study, using a time-mapping approach, the eigenvalue-like problem (phi(u'))' + A phi(u(+)) - B phi(u(-)) = 0, u(a) = 0 = u(b), where A > 0, B > 0. We show here that this PFS, which we denote by P, consists of a set of curves in (R(+))(2) resembling the classical Fucik spectrum, i.e., when phi(s) = s. Our main existence result, which deals with nonresonance with respect to the PFS, can be roughly stated as follows: if for s > 0 sufficiently large and for almost every t is an element of [a, b], the pair (f(t, - s)/phi(- s>, f(t, s)/phi(s)) lies in a compact rectangle contained in an open component of (R(+))(2)\P which intersects the diagonal, then problem (P) has at least a solution.

[发布日期] 1994-07-20

[关键词] NONRESONANCE;TIME MAPPING;PSEUDO-EIGENVALUE;PSEUDO FUCIK SPECTRUM;CONTINUATION LEMMA