UNIQUENESS OF POSITIVE RADIAL SOLUTIONS OF A SEMILINEAR ELLIPTIC EQUATION IN AN ANNULUS

. In this paper, we show the following equation on ∂ , has at most one positive radial solution for a certain range of λ > 0. Here p > 1 and Ω is the annulus { x ∈ R n : a < | x | < b } , 0 < a < b . We also show this solution is radially non-degenerate via the bifurcation methods.


1.
Introduction. This paper is concerned with the uniqueness of the positive radial solution of the Dirichlet problem of ∆u + f (u) = 0 (1) in the annulus Ω = {x ∈ R n : a < |x| < b} where 0 < a < b.
Problem (1) serves as a model in many different areas of applied mathematics and has been extensively studied by several mathematicians in the last three decades.
When Ω is a ball or the entire space, it is well known [5,6] that all positive Dirichlet solutions of (1) must be radially symmetric (up to translation), provided certain conditions on f hold. Many authors focus on the uniqueness of positive solutions in a finite ball or the entire space R n ; see [10,13,14,16,18,22] and the references therein, while many exact multiplicity results have been obtained, see [12,17] and the references therein.
When Ω is an annulus, the Dirichlet problem is ∆u + f (u) = 0 for a < |x| < b, u = 0 for |x| = a, b, where 0 < a < b. The positive solution of (2) is not necessarily radially symmetric, and some results of the nonradial solutions have been obtained in [1,8,15]. While for some subcritical nonlinearities and for annulus with small interior radius, it has been proved in [9] that all positive solutions of (2) are radial. For f (u) = u p ± u, the uniqueness of positive radial solutions has been studied by [2,4,16,20,21]. The aim of this paper is to give some uniqueness results for the following problem      ∆u + u p + λu = 0 for a < |x| < b, u = 0 for |x| = a, b, u > 0 for a < |x| < b where 0 < a < b < ∞, n ≥ 2, p > 1 and λ > 0. Ni & Nussbaum [16] have shown the uniqueness results of radial solutions under one of the following conditions: (ii) the annulus is thin, say, 1 < b/a < c n where c n = (n − 1) 1/(n−2) for n ≥ 3, and c 2 = e for n = 2.
(iii) λ = 0 (in the special case, there is no restriction on p and the rate of b/a).
In fact, in [16], (i)-(iii) have been proved for a more general nonlinearity than the one mentioned here. The condition (ii) has been improved by Korman [12], in which he shows the uniqueness results for 1 < b/a ≤ c n where c n = (2n − 3) 1/(n−2) for n ≥ 3, and c 2 = e 2 for n = 2. In [21], Yadava shows the uniqueness results for the subcritical and critical case 1 < p ≤ (n + 2)/(n − 2).
For 0 < λ < λ 1 (a, b), the least energy solution (restricted in the radial space) always exists, which is a positive radial solution. Here λ 1 (a, b) denotes the first Dirichlet eigenvalue of −∆ in the annulus {x ∈ R n : a < |x| < b}. For λ = 0 the radial solution of (3) is unique and radially non-degenerate (see [16]). So, it is also unique for a sufficiently small and positive λ by the implicity function theorem. We show that the radial solution of (3) is also unique for a certain range of λ.
(ii) p > (n + 2)/(n − 2), n > 2 and λ ∈ (0, Here β and L are defined in (5) with ν = n − 1. Then the Dirichlet problem (3) has at most one radial solution. Moreover, this unique radial solution is non-degenerate in the space of the radially symmetric function.
In [21], they focus on the non-degeneracy of the solution, while we mainly establish the uniqueness result first and then the non-degeneracy. We mainly focus on the super-critical case and our methods can be also applied for the subcritical and critical case, provided another proof of the result by Yadava [21].
If n ≥ 3, p > (n + 2)/(n − 2), then β > 2, L > 0, and G is decreasing-increasing in (0, ∞) with the unique minimum point r =r = (β − 2)L/(λβ). Ifr ∈ (a, b), The proof of our theorem is based on the ideas developed in [14], [11] and [4]. In section 2, we proceed to prove that the solution is unique by a contradiction argument, assuming there is more than one solution. In doing so, we first characterize two possible positive solutions by the number of crossing points and then we use an energy analysis to get a contradiction. In section 3, we prove the non-degeneracy of the positive solution by a contradiction argument, assuming the positive solution ϕ is degenerate. We construct a perturbed equation, with ϕ as a trivial solution, and use the bifurcation methods to get a contradiction.
2. The uniqueness of positive radial solutions. In this section, we will prove the uniqueness part of Theorem 1.2, that is, (4) has at most one solution.
Before proving the uniqueness result, we point out the existence of solutions of (4). It is clear that a necessary condition for the existence is V satisfiesλ 1 On the other hand, the solution of (4) is a critical point of the functional for u ∈ H 1 0 (a, b). If V satisfiesλ 1 (V ) > 0, then the functional J has the mountain pass structure and there exists a mountain pass solution [19], which is a solution of (4).
(1) Any two distinct solutions u 1 , u 2 of (4) must intersect at least once in (a, b).
It immediately follows that the interior zero of ξ is just the intersection points of u 1 (r) and u 2 (r). Observing that ξ(a) = ξ(b) = 0, we know ξ = 0 has at least one interior zero, that is, u 1 (r) and u 2 (r) intersect at least once in (a, b).
If the solution of problem (4) is not unique, then we want to find two solutions u 1 , u 2 which intersect exactly once, i.e., u 1 , u 2 satisfies (8). In order to do this, we will use the solution of the initial value problem Let u = u(r; γ) be the solution of (10) and let R = R(γ) be the first zero of u(·; γ), i.e., R(γ) = sup{R > a : u(r; γ) > 0, r ∈ (a, R )}.
Proof. Let c 1 ∈ (a, b) be the first critical point of u 1 and M 1 = u 1 (c 1 ). From Lemma 2.2, there exists a large constant γ 1 such that for every γ ≥ γ 1 , R(γ) < c 1 and the function u = u(r; γ) has a unique critical point (denoted by c = c(γ)) in (a, R(γ)).

Now we set
Then w satisfies where β and L are constants given in (5). Denote and Then Proof. Suppose that (4) has two distinct solutions, then by Lemma 2.4, Lemma 2.1, we know (4) has two distinct solutions u 1 , u 2 satisfying (8) and (9). Now we set for r ∈ (a, b) by (9) and Lemma 2.5.
On the other hand, we have that This is a contradiction. This completes the proof.
Remark 2. The condition (V2) is used just to guarantee the positivity of E(r, w) in Lemma 2.5.
3. The non-degeneracy of positive radial solutions. In this section we prove the non-degeneracy of the unique solution by a contradiction argument based on the bifurcation theory of a perturbed equation. We say the solution u of (4) is non-degenerate if the linearized problem does not admit any non-trivial solution. Let us denote by ϕ = ϕ(r) the unique positive solution of (4). For any δ ∈ R, we consider a perturbed equation related to (4) Clearly, u = ϕ is always a solution of (19). We first prove a uniform bounded result by the blow up methods.
Lemma 3.1. Let Λ > 0 be any fixed constant. Then there exists a constant K depending only on Λ such that if u is a solution of (4) with |V (r)| < Λ then Proof. The proof is based on the blow up methods [7]. We argue by contradiction and suppose that there exists a sequence of solutions u k (related to Without loss of generality, we assume thatb k → ∞. Note that for any R > 0, v k C 2 [0,R] is bounded as k → ∞. By passing subsequence if necessary, we may assume that v k converges in C 1+1/2 loc [0, ∞) to a function v, and v is a weak (then classical) solution to Here we use the fact M −(p−1)/2 k s + r k > a > 0. Since v ≡ 1, we know there is a s 1 such that v s (s 1 ) < 0, and hence v s (s) ≤ v s (s 1 ) < 0 for s > s 1 , this leads v to be unbounded from below, a contradiction.
The corresponding "energy" to (19) is denoted by E δ = E δ (r, u) as follows (20) Now we turn to prove the uniqueness of positive solutions of (19).
Proof. In order to derive the non-degeneracy, we argue indirectly. Suppose that this unique solution ϕ of (4) is degenerate. Noting that u = ϕ is always a solution of (19), we can use the local bifurcation theorem (See [3]) to obtain another solution which will be a contradiction. Denote by F : ] the map as follows where δ is the bifurcation parameter, C 2 0 [a, b] = {u ∈ C 2 [a, b] : u(a) = u(b) = 0}. Note that F is C 2 . It is easy to calculate the Frechlet derivatives at (u, δ) = (ϕ, 0), L 0 = F u (ϕ, 0) = ∂ rr + ν r ∂ r + (pϕ p−1 + V ), and F uδ (ϕ, 0) = (p − 1)ϕ p−1 . By the degeneracy of ϕ, we know 0 is a simple eigenvalue of L 0 . It is clear that the kernel of L 0 is spanned by some function φ (φ ≡ 0) and the range space of L is characterized by R(L 0 ) = {h ∈ C[a, b] : b a φ(r)h(r)r ν dr = 0}.
It immediately follows that for some s (with |s| small), F has a non-trivial zero, which is another positive solution of (19). This contradicts the uniqueness proved in Lemma 3.2. This completes the proof.