One-dimensional p-Laplacian with a strong singular indefinite weight, I.: Eigenvalue

In this paper, we prove the existence of eigenvalues for the problem {phi(p)(u'(t))' + lambda h(t)phi(p)(u(t)) = 0, a.e. in (0, 1), u(0) = u(1) = 0, where phi(p)(s) = |s|(p-2)s, p > 1, lambda is a real parameter and the indefinite weight h is a nonnegative measurable function on (0, 1) which may be singular at 0 and/or 1, and h not equivalent to 0 on any compact subinterval in (0, 1). We derive similar properties of eigenvalues as known in linear case (p = 2) or continuous case (h is an element of C[0, 1]) if h satisfies integral(1)(0) t(p-1) (1-t)(p-1) h(t) dt < infinity when 1 < p <= 2 and integral(1/2)(0) phi(-1)(p)(integral(1/2)(s) h(tau)d tau)ds + integral(1)(1/2) phi(-1)(p) (integral(s)(1/2) h(tau)d tau)ds < infinity when p >= 2, respectively. For the result, we establish the C-1-regularity of all solutions at the boundary for the above problem as well as the following problem: {phi(p)(u'(t))' + lambda h(t)f(u(t)) = 0, a.e. in (0, 1), u(0) = u(1) = 0, where f is an element of C (R, R), sf(s) > 0 for s not equal 0, f is odd and f(s)/phi(p)(s) is bounded above as s -> 0. (C) 2007 Elsevier Inc. All rights reserved.