ANALYSIS OF POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE p -LAPLACIAN PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS

. We study positive solutions to (singular) boundary value problems of the form: 0 , where ϕ p ( u ) := | u | p − 2 u with p > 1 is the p -Laplacian operator of u , λ > 0, 0 ≤ α < 1, c : [0 , ∞ ) → (0 , ∞ ) is continuous and h : (0 , 1) → (0 , ∞ ) is continuous and integrable. We assume that f ∈ C [0 , ∞ ) is such that f (0) < 0, lim s →∞ f ( s ) = ∞ and f ( s ) s α has a p -sublinear growth at inﬁnity, namely, lim s →∞ f ( s ) s p − 1+ α = 0. We will discuss nonexistence results for λ ≈ 0, and existence and uniqueness results for λ (cid:29) 1. We establish the existence result by a method of sub-supersolutions and the uniqueness result by establishing growth estimates

In the case when f (0) < 0 and α = 0, (1) is referred to as an infinite semipositone problem. The study of positive solutions to these problems is very challenging since ranges of positive solutions must include regions where f is negative as well as where f is positive.
The boundary value problem (1) arises in the study for radially symmetric steady states of reaction diffusion equations of the form: where ∆ p u := div(|∇u| p−2 ∇u), 1 < p < N , Ω = {x ∈ R N | |x| > r 0 > 0} and ∂u ∂η is the outward normal derivative of u on |x| = r 0 . Here for the case when c(u) is a positive constant (Robin boundary condition case) there is a rich history of results (see [1], [5] and [11]), while this is not the case when c(u) is not a constant. The case when c(u) is not a constant occurs naturally in various applications, see [6], [14] and [16] where they discuss models arising in chemical reaction theory, and see [3], [4] and [7] where they discuss models arising in population dynamics. In particular, in population dynamics, the case where c(u) is not a constant occurs when species exhibit strong density dependent behavior at habitat boundaries. Restricting the analysis to positive radial solutions, by a Kelvin type transformation, namely the change of variable r = |x| and t = r r0 N −p 1−p , (2) reduces to analyzing the two point boundary value problem (1).
We first establish the following nonexistence result: There exists no positive solution of (1) for λ ≈ 0.
Finally, we assume: and establish the following uniqueness result: When p = 2 and α = 0, the authors in [2] established the nonexistence of a positive solution for λ ≈ 0 and the existence result for λ 1. In [10], the authors extended these results to the case when p = 2 and α = 0. Theorems 1.1 -1.2 are extensions of these results in [2] and [10] to the case p > 1. Further, in [9], the uniqueness result for λ 1 was established when p = 2 and α = 0. Theorem 1.3 is the extension of this uniqueness result to the case when p > 1 and α = 0. These extensions to the case p = 2 are nontrivial and very challenging due to the presence of the nonlinear p-Laplacian operator. Further, we do not require the concavity assumption on f as in [9], instead, we use the weaker condition (H 6 ).
2. The method of sub-supersolutions. For a subsolution ψ and a supersolution φ such that ψ ≤ φ, we define the operator T : where γ : [0, 1] × R → R andc : R → R are defiend by It follows that T satisfies the following properties: Next we show that T is continuous. Let {w n } ⊂ C[0, 1] be such that w n → w as n → ∞ for some w ∈ C[0, 1]. Sincec is continuous, ϕ p (c(w n (1))) converges to ϕ p (c(w(1))) as n → ∞. We also have Since the last term converges to 0 as n → ∞ by the Lebesgue Dominated Convergence Theorem, γ(s,wn) α ds converges uniformly to 1 t h(s) f (γ(s,w)) γ(s,w) α ds as n → ∞. For each n and t ∈ [0, 1], we have For t ∈ [0, 1], we also have , we obtain that T w n (t) converges uniformly to T w(t) as n → ∞. This implies that T w n − T w ∞ → 0 as n → ∞, so T is continuous. Hence Lemma 2.1 is proven.
This implies that there exists M * 1 such that (I − T )(w) = 0 for any w ∈ C[0, 1] satisfying w ∞ = M * . Then, by the Homotopy Invariance Theorem, we have 1]. This implies that w 0 is a positive solution of (1), and hence Lemma 1.4 is proven.
Assume that u is a positive solution of (1). Then we have . This is a contradiction for λ ≈ 0. Hence there exists no positive solution of (1) for λ ≈ 0.

This implies that
Thus we obtain By Lemma 4.1, for λ 1 there exists C 1 > 0 such that Hence, for λ 1 we have Next we show that there exists where u 1 := u ∞ + u ∞ . By Lemma 4.1, for λ 1 there exists C 2 > 0 such that Thus we have u(t) ≤ u ∞ t ≤ C 2 G −1 (λ 1 p−1 )t for λ 1. Hence the proof is complete. Now we recall Lemma 2.9 in [15] which we restate as Lemma 4.3 below. We also provide a proof since there was an error in the arguments in [15].