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摘要
In this short note, we establish the following result: Let f: [0, +infinity[-> [0, +infinity[, alpha : [0,1]->]0, +infinity[ be two continuous functions, with f(0) = 0. Assume that, for some a > 0, the function xi -> integral(xi)(0) f(t)dt/xi(2) is non-increasing in ]0, a].
Then, the following assertions are equivalent:
(i) for each b > 0, the function xi -> integral(xi)(0) f(t)dt/xi(2) is not constant in ]0, b];
(ii) for each r > 0, there exists an open interval I subset of]0, +infinity[such that, for every lambda is an element of I, the problem
{-u '' = lambda alpha(t)f(u) in [0, 1]
u > 0 in ]0, 1[
u(0) = u(1) = 0}
has a solution u satisfying
integral(1)(0)vertical bar u'(t)vertical bar(2)dt<r.
- 出版日期2015
