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For critical case, a positive solution was obtained in [14] with by the Lagrange multipliers technique.

Nonexistence of a positive solution was also proved when a ≤ λ 1.

Moreover, the nonexistence of a positive solution is also studied.

In this paper, a positive solution is a function u ∈ C [ 0, 1 ] ∩ C 2 ( 0, 1 ) satisfying (1.3).

Furthermore, the interval of α about the nonexistence of a positive solution is also given.

Applying Theorem 2.1, the following new result on the existence of a positive solution is obtained.

The existence and uniqueness of a positive solution are given in Section 2.

As we mentioned above, BVP (1.1) having a positive solution is equivalent to BVP (2.2) having a positive solution.

The existence of a positive solution is guaranteed for large λ by applying the sub-supersolution method.

The existence of a positive solution is a consequence of the fixed point theorem (see Deimling [16], Theorem 20.1) by virtue of Lemma 3.1 and Lemma 3.2.

Generalizing some results from [2], the existence of a positive solution is proved with Brouwer's theorem, and the uniqueness is a consequence of suitable inequalities.