We investigate the number of positive solutions of the semilinear boundary value problem{A formula is presented}where BR ⊂ Rn is the ball centered at the origin with radius R, and f : ( 0, ∞ ) → ( 0, ∞ ) is a locally Lipschitz continuous function satisfying the following conditions:(H1) f ( u ) {greater than or slanted equal to} m > 0 ( u > 0 ) ;(H2)there exists {A formula is presented}where 2* = frac(n + 2, n - 2) if n > 2 and 2* = ∞ if n {less-than or slanted equal to} 2.(H3){A formula is presented}The motivating example, involving singular nonlinearity, is the problem{A formula is presented}with some constants {A formula is presented}This problem can be reduced to (1) with f ( s ) = s- α + sp using a suitable transformation. Problems with singular nonlinearities have been studied extensively recently involving more general nonlinearities which may also depend on x. For the case of singularities in x, see [10,13] and references therein. We are mainly interested here in singularities in u. For nonlinearities where the term up does not appear, existence and uniqueness has been proved in [3,5,9,11,12,17]. For equations of the form Δ u + frac(λ, uα) - up, existence and uniqueness for p > 0 (and also for p = - β < 0, β < α) can be found in [12]. (For p = - β < 0, α < β < 1 non-existence is expected [12].). In the presence of up the problem exhibits a different behavior according to the sign of λ. If λ < 0, then there may be no positive solution; existence and non-existence results for the concave case p ∈ [ 0, 1 ) can be found in [6,21,24] and generalized for x-dependent nonlinearities in [4,12]. In the latter case non-existence of positive solutions under the critical value λ* still allows existence of non-negative solutions with a free boundary (see [4]). Related results can be found in [20] for the nonlinearity λ ( up - u- α ). When the parameter λ multiplies the term up, then existence and uniqueness holds for all λ if u- α has a negative coefficient [12]. Here non-existence can also occur (namely for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0) when u- α has a changing-sign coefficient or when the term up is replaced by a bounded non-negative function f ( x ) (see [6,12]). Conversely, a unique positive solution exists exactly for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0 in two related cases when the equation contains three terms: Δ u + λ u + f ( x ) equals u- α or - u- α. For p = 0 there are also some multiplicity results in [1]. For λ > 0, existence is easier to prove on a ball and has also been obtained on smooth bounded domains. For p ∈ ( 0, 1 ) there is always a positive solution, which is also unique [12,21]. On the other hand, for p > 1 existence only holds for small λ [2,12]. Existence of a second positive solution has been proved for associated non-singular problems by using variational arguments. Some related results for different concave and/or convex nonlinearities can be found in [14-16,19]. For the singular case we prove here the existence of a second positive solution in the case of a ball and, moreover, that there are exactly two radial positive solutions for n = 1. At the best of our knowledge, the celebrated result by Gidas et al. [8] has not been extended to the case of singular nonlinearities. Hence we cannot be sure that our problems have only radial solutions in the case of a ball, and consequently our results for the case n > 1 refer only to positive radial solutions. The main results of our paper are formulated as follows. {A formulation is presented}. In order to have exact multiplicity for n = 1, f is assumed to satisfy a further condition:(H4) f is a strictly convex C2 function(i.e. f″ {greater than or slanted equal to} 0 and f″ does not vanish identically on any interval).{A formulation is presented}. Concerning the main example (2), Theorems 1 and 2 yield the following result using the transformation {A formula is presented}{A formulation is presented}. The proof of our results relies on the shooting method. Theorem 1 is proved in Section 3 via characterizing the shape of the time-map. The exact multiplicity is dealt with in Section 4 in a little more general setting than in Theorem 2, namely, we assume a certain admissibility condition (H5), which is seen to be always satisfied in the case n = 1. © 2006 Elsevier Ltd. All rights reserved.
