Uniqueness and asymptotic behavior of solutions of a biharmonic equation with supercritical exponent

Existence and uniqueness of positive radial solution \begin{document}$ u_p $\end{document} of the Navier boundary value problem: \begin{document}$ \left \{ \begin{array}{ll} \Delta^2 u = u^p \;\;\; &\mbox{in $ \mathbb{R} ^N \backslash {\overline B}$},\\ u>0 \;\;\; &\mbox{in $ \mathbb{R} ^N \backslash {\overline B}$},\\ u = \Delta u = 0 \;\;\; &\mbox{on $\partial B$}, \end{array} \right. $\end{document} where \begin{document}$ B \subset \mathbb{R} ^N \; (N \geq 5) $\end{document} is the unit ball and \begin{document}$ p>\frac{N+4}{N-4} $\end{document} , are obtained. Meanwhile, the asymptotic behavior as \begin{document}$ p \to \infty $\end{document} of \begin{document}$ u_p $\end{document} is studied. We also find the conditions such that \begin{document}$ u_p $\end{document} is non-degenerate.

where B ⊂ R N (N ≥ 5) is the unit ball and p > N +4 N −4 , are obtained. Meanwhile, the asymptotic behavior as p → ∞ of up is studied. We also find the conditions such that up is non-degenerate.
1. Introduction. We consider existence and uniqueness of positive radial solutions of the Navier boundary value problem: where B ⊂ R N (N ≥ 5) is the unit ball, i.e., B = {x ∈ R N : |x| < 1} and p > N +4 N −4 . The structure of positive solutions of the equation is considered by many authors recently, see [1,3,6,9,10,12,13,15,16,17,19,23,25]. The classification of positive entire solutions of (1) via Morse index has also been obtained, see [5,21,23,24,30]. In the supercritical case, i.e., when p > N +4 N −4 , there are no positive solutions of the Navier boundary value problem: when Ω is star-shaped or a starlike domain (see [27,29]). On the other hand, the existence results of positive solutions have been established when Ω is topologically nontrivial in the spirit of Bahri-Coron (see [2,8]) or when it is contractible with some special geometry (see [11]). We are able to say that both topology and geometry of the domain Ω play important roles in the existence of solutions for (2). In this paper, we establish the existence and uniqueness of positive radial solution u p of (P) and obtain the asymptotic behavior as p → ∞. To obtain the existence and uniqueness of u p , by using the Kelvin transformation, we need to consider existence and uniqueness of positive radial solution v p,α * of the problem: where α * = (N − 4)p − (N + 4) > 0. The existence and uniqueness of the positive least energy radial solution w p of the problem   with p > N +4 N −4 are obtained in [20], which was used to construct nontrivial solutions to the problem where B (x 0 ) ⊂⊂ Ω and p > N +4 N −4 . A variant of the arguments in [7,20] implies that we can also construct nontrivial solutions to the problem where B (x 0 ) ⊂⊂ Ω and p > N +4 N −4 , if we understand the properties u p of (P), in particular, the non-degeneracy of u p . We can show that the unique positive least energy radial solution obtained in [20] is actually the unique positive radial solution to (4). Indeed, we can show that the equation in (3) with the boundary conditions: v = ∂v ∂ν = 0 on ∂B admits a unique positive radial solution by using a variant of the arguments in [4]. The main purpose of this paper is the uniqueness of u p , the asymptotic behavior of u p as p → ∞ and the conditions such that u p is nondegenerate. Meanwhile, we also provide the asymptotic behavior (as p → ∞) of the unique positive radial solution w p of (4).
Our main results of this paper are the following propositions and theorems. with where does not admit any nontrivial radial solution.
Theorem 1.4. The following estimate holds Remark 1.5. The consequence of Theorem 1.4 provides an important difference between the fourth order case and the second order case. In [14], by using wp p+1 ∞ p+1 → D > 0 as p → ∞, where w p is the unique positive radial solution of the Dirichlet problem: −∆w = w p in Ω, w = 0 on ∂Ω with Ω being an annulus, the author obtains the relation between w p and a solution of an initial value problem of a second order ODE with an exponential nonlinearity via a blow-up argument. The authors in [7] mentioned that the similar behavior also holds for Ω = R N \B. We can not use the similar arguments to our u p here since the limit as p → ∞ of up p+1 ∞ p+1 is 0.
The organization of this paper is as follows. In section 2 we establish the existence and uniqueness of positive radial solution u p of (P). In section 3, we obtain the asymptotic behavior of u p as p → ∞ and provide the proof of Theorems 1.2 and 1.4. In the final section, we find the conditions such that u p is non-degenerate. In this paper, we use C to denote a universal positive constant, which may change from one line to another line.

2.
Existence and uniqueness of positive radial solution u p of (P). In this section we mainly demonstrate the existence and uniqueness of positive radial solution u p of (P). To do this, we first obtain the existence and uniqueness of positive radial solution of the problem (3) with α * being replaced by a nonnegative number α.
The existence and uniqueness of positive radial solution u p of (P) can be obtained from Proposition 2.1 and the Kelvin transformation.
Proof of Proposition 2.1. To obtain the existence of v p,α , we consider This implies that the norm of H 2 rad (B) is equivalent to It is known from [18] that the embedding: is compact for 1 ≤ p < N +4+2α N −4 . Meanwhile, choosing V (x) ≡ 1 and W (x) ≡ 0, it follows from (3) of Theorem 1.1 in [28] (Note that the number θ in [28] is 0.) This implies that, for N ≥ 5, Both (12) and (13) imply that A p,α is attained at some v p,α ∈ H 2 rad (B) and (12) implies that A p,α > 0. Note that we see from (14) We claim that v p,α is nonnegative. On the contrary, we consider the solution of −∆w = |∆v p,α | in B, w = 0 on ∂B.
We now show the uniqueness of v p,α . We will see that we can not even obtain our uniqueness of the least energy radial solution by using arguments of [20] here. In [20], by the Pohozaev's identity, the authors notice that if z p (ρ) := r N −4 w p (r) and ρ = r −1 with w p being a positive least energy radial solution of (4), then z p (1) depends only on p. Since the boundary conditions of (11) is a little more complicated than that of [20] (note that the boundary conditions in [20] are z p (1) = z p (1) = 0), if we use the similar arguments in v p (ρ) := r N −4 u p (r) with ρ = r −1 , where u p is a positive least energy radial solution of (P), we can obtain the related Pohozaev's identity and that 2(4 − N )(v p (1)) 2 − v p (1)(∆v p ) (1) depends only on p, which can not be used to obtain our uniqueness of by arguments as in [20]. So, we will use different arguments to obtain our uniqueness here. Suppose by contradiction that the problem (11) admits two different positive radial solutions v and v (we omit the subscripts p and α). Let λ 4+α p−1 = v(0) v(0) . We define the function: Moreover, we can also see that We now show that (∆v)(0) = (∆w)(0).
where C is independent of p.
Proof. Arguments similar to those in the proof of Proposition 2.1 imply that the eigenvalue problem: admits an eigenvalue σ 1 > 0 and an eigenfunction ϕ 1 > 0 corresponding to σ 1 .
Multiplying ϕ 1 on both the sides of the equation of v p,α and integrating it on B, we see that This implies that |x| Since α ≥ 0, we see from (29) that This implies that (24) holds. We can easily obtain (25) from (24) by a simple contradiction argument. We now show (26). Note that α * > 0 implies that p > N +4 On the other hand, it follows from (41) of Proposition 3.4 in next section and Arzela-Ascoli's theorem that there isv such that (up to a subsequence) v pi,α i * →v uniformly on [0, 1] as i → ∞. This and (31) imply thatv ≡ 0 in [0, 1]. This contradicts (24). This contradiction implies that (26) holds.
This implies that 1 < p < N +4+2α * be a solution to (P). Making the Kelvin transformation:

ZONGMING GUO, XIAOHONG GUAN AND YONGGANG ZHAO
we know from Lemma 3.1 of [19] that v(y) satisfies the problem , it follows from Proposition 2.1 that (33) admits a positive The uniqueness of v p,α * implies the uniqueness of u p . This completes the proof of this theorem.
3. Asymptotic behavior of u p as p → ∞: Proof of Theorems 1.2 and 1.4.
In this section, we study the asymptotic behavior of u p obtained in Theorem 1.1 as p → ∞. The asymptotic behaviour of the solution as p → ∞ of the second semilinear elliptic problem where Ω is an annulus of R N , N ≥ 2 is studied in [14]. Let We have the following proposition.
Then we have that for p > N +4 N −4 sufficiently large, where r 0 is given in Theorem 1.2.
Proof. Both the results of Proposition 2.1 and the Kelvin transformation imply that I p attains at some nonnegativeũ p ∈ G. Note that a simple calculation implies that, for any ψ ∈ G, where ψ(r) = s N −4 φ(s), r = 1 s and for any φ ∈ H 2 rad (B), [13,28]), Considering the function (|x|) given in (8), we see that ∈ G and ∞ := max [0,∞) (r) = 1 (this also applies in the following). It follows from the definition of I p that This completes the proof of this proposition.
Corollary 3.2. Let u p be the unique positive radial solution of (P). Then we have that for p sufficiently large, where C is a positive constant independent of p.
Proof. Let us consider a minimizerũ p to I p . We have thatũ p solves the problem Since the radial solution u p to (P) is unique, we derive that u p = I  Proof. We have that u p satisfies From the equation of u p , we easily see that there is exactly one r p ∈ (1, ∞) such that u p (r p ) = 0 and r p is the maximum point of u p . Indeed, suppose that u p admits a local maximum point and a local minimum point in (1, ∞), then u p has another maximum point and there are r 1 p , r 2 p ∈ (1, ∞) such that (∆u p ) (r 1 p ) = (∆u p ) (r 2 p ) = 0. Integrating the equation of u p in (r 1 p , r 2 p ), we derive a contradiction. Similar arguments imply that there is exactly one r * p ∈ (1, ∞) such that (∆u p ) (r * p ) = 0. Note that r p and r * p are the unique maximum and minimum point of u p and ∆u p in (1, ∞) respectively. Integrating (39) in (r, r * p ) ⊂ (1, r * p ) and (r * p , r) ⊂ (r * p , ∞), we obtain The following proposition mainly present the uniform boundedness of ∇(∆v p,α * ) for any p > N +4 N −4 . We would like to point out that this result can not be obtained directly by arguments similar to those in the proof of Proposition 3.3.
where C is a constant independent of p. Moreover, where C > 0 is independent of p. Furthermore, for |x| sufficiently large, where C is independent of p.
Proof. Let˜ We see that˜ ∈ H 2 rad (B). Using˜ as the test function, arguments similar to those in the proof of Proposition 3.1 imply that where C > 0 is independent of p. It follows from (14) and the embeddings that where C > 0 is independent of p. To see this, we notice that v p,α * = A 1 p−1 p,α * v p,α * and We also know that It is known from (14) that for any sufficiently small δ > 0, v p,α * ∈ H 2 (1 − δ, 1). Moreover, the embedding where C > 0 is independent of δ and p. Since we can choose δ such that we see from (47)-(49) that where C > 0 is independent of p. We obtain (45) from (46) since A 2 p−1 p,α * ≤ C and C > 0 is independent of p.
We now claim that there are > 0 sufficiently small and C > 0 independent of p such that, for any p > N +4 To see this, we notice from the embedding and (45) that there is C > 0 independent of p such that Using the fact that v p,α * (r) is decreasing, we have that for any > 0 and r ∈ (0, ], This and (54) imply that for r ∈ (0, ], where C > 0 is independent of p. Similar arguments imply that for r ∈ (0, ], where C > 0 is independent of p.
Proof. It follows from (26) and u p (s) = s 4−N v p,α * (1/s) that where C > 0 is independent of p. Therefore, is independent of p. This implies that (64) holds. Both (42) and (64) imply that there is M > 0 independent of p such that where r p and r * p , defined in the proof of Proposition 3.3, are the unique maximum and minimum point of u p and ∆u p in (1, ∞), respectively.
To obtain (65), we also use contradiction arguments. Suppose that there is a sequence Since (67) implies that 1 < r pi ≤ M , a simple calculation gives that This contradicts (64) and completes the proof of this lemma.
Corollary 3.6. We have Proof. Let r * p be defined as in the proof of Proposition 3.3. We integrate (∆u p ) (r) in (1, r * p ) and obtain from (38) that where C > 0 is independent of p. Thus Similar arguments imply |u p | ≤ C, |u p | ≤ C.
From (38), (70), (71) and Ascoli-Arzela's theorem we obtain that u p → u in C 2 loc [1, ∞) as p → ∞. By Lemma 3.5, we derive that u ≡ 0 and ∆u ≡ 0. Let The boundary conditions of u p and ∆u p and the conclusions in Lemma 3.5 imply Proof. Let us consider the case r < r 0 (the proof of the case r > r 0 is the same). By contradiction, let us suppose that there existr < r 0 and a sequence {p i } with p i → ∞ as i → ∞ such that u pi (r) ≥ 1.
Since u pi is strictly increasing in [1, r pi ] we derive that u pi (r) > 1 for any r ∈ (r, r pi ]. Let us denote by r * pi the unique minimum point of ∆u pi and r * 0 = lim i→∞ r * pi . We consider three cases here: (i) r * 0 < r 0 , (ii) r * 0 = r 0 , (iii) r * 0 > r 0 .
We only consider the first case, other two cases can be studied similarly. We integrate the equation of u pi in (r * pi , r pi ) and obtain that no matter r * 0 ≤r or r * 0 >r. This contradicts to (38). For the other two cases, we integrate the equation of u pi in (r, r * pi ) respectively, we can also derive contradictions.
Corollary 3.8. Let r p and r * p be the maximum and minimum points of u p and ∆u p respectively. We have Suppose that r 0 = r * 0 . There are two cases: (a) r 0 > r * 0 , (b) r 0 < r * 0 . For the first case, it follows from Lemma 3.7 that u p (r) < 1 for r ∈ (1, r * p ). Therefore, for any sufficiently small > 0, there isp 0 > 1 such that for p >p 0 , This contradicts to (65). For the second case, we first notice that we can choose R * > r * 0 such that |(∆u p )(R * )| < C 10 , where C is given in (65). It follows from Lemma 3.7 that u p (r) < 1 for r ∈ (r * p , R * ). Therefore, for any sufficiently small > 0, there is p 1 >p 0 > 1 such that for p >p 1 , (r N −1 (∆u p ) (r)) < ∀r ∈ (r * p , R * ). Integrations imply that, if we choose > 0 sufficiently small, This contradicts to (65).
Proof of Theorem 1.2. It follows from Corollary 3.6 and Lemma 3.9 that u satisfies A straightforward computation shows that u ≡ . This completes the proof of this theorem.
Remark 3.10. Arguments similar to those in the proof of Theorem 1.2 and simple calculations imply that the following conclusion holds (note that (∆w p )(1) > 0 and (∆w p ) (1) < 0): Let w p be the unique positive radial solution of (4). Then, as p → ∞, with where The numbers m 1 > 0, m 2 < 0 andr 0 > 1 can be determined by the following system of equations: Proof of Theorem 1.4. Multiplying u p on both the sides of the equation and integrating it on (r p , ∞), we see that Therefore, via a simple calculation.
Remark 3.11. We can obtain the following conclusion by using Remark 3.10 and arguments similar to those in the proof of Theorem 1.4. Let w p be the unique positive radial solution of (4). Then, the following estimate holds 4. Non-degeneracy of u p . The purpose of this section is to see that under what conditions the problem admits only the trivial solution φ ≡ 0. We first study the problem This eigenvalue is characterized as The number ν(p) is negative, since this Rayleigh quotient gets negative when evaluated at ψ = u p . This fact, the Hardy's inequality and a simple compactness argument involving the fast decay u p−1 p = o(r −4 ) at r = ∞, yield the existence of an extremal for ν(p) which represents a positive solution to (84) for ν = ν(p).
We first present the proof of Proposition 1.3.
Proof of Proposition 1.3. To obtain the conclusion of Proposition 1.3, we first consider the function ρ p (r) := ru p (r) + 4 p−1 u p (r). For convenience, we omit the subscript p of ρ p in the following. It is easily seen via a simple calculation that ρ satisfies the equation We now claim that there isr p ∈ (1, ∞) such that We see that (∆ρ)(r) = ∆(ru p (r)) + 4 p − 1 (∆u p )(r) = r(∆u p ) (r) + 2(p + 1) p − 1 (∆u p )(r).
As in the previous section let u p be the unique positive radial solution of (P). We want to analyze the possible degeneracy of the linearized operator L up = ∆ 2 − pu p−1 p I. To this aim let us denote by β a generic eigenvalue of the problem We will find the conditions such that β = 0.
Proof. By the Kelvin transformation, we can change the eigenvalue problem (96) to a related eigenvalue problem on B. We see that whereβ Then (97) can be easily obtained from (99) and the expression ofβ k . To prove (98), we only need to show We see that β Since u p is a mountain pass solution, the radial Morse index of u p is at most 1, we have that β (2) 0 ≥ 0. To see this, let us suppose that the operatorL p,r := r 4 (∆ 2 −pu p−1 p I) admits at least 2 negative (radial) eigenvalues, is simple. We assume that ϕ k are the associated eigenfunctions ofL p,r . Since ϕ 1 is orthogonal to ϕ 2 in L 2 rad (R N \B, |x| −4 ), we have that From this we see that is negative on X := {sϕ 1 + tϕ 2 : s, t ∈ R} except at the origin and hence the radial Morse index of u p is at least two. This contradicts the fact that the radial Morse index of u p is at most 1. Therefore, β We now have the following theorem.
Then u p is non-degenerate.
Proof. It follows from (98) that β (i) k = 0 for k ≥ 1 and i ≥ 2. The only possibility for β k = 0 is β (1) k = 0. Assume that p is such that (101) holds, then problem (83) admits only the solution φ ≡ 0 and u p is non-degenerate.
It is remarkable to note that β where σ (1) k (p) is the first eigenvalue of (105). Then u p is nondegenerate. Using arguments similar to those in the proof of Proposition 1.3 to (104), we can obtain that σ  k (p) = p} is finite. We use arguments similar to those in the proof of Proposition 3.5 of [20]. We first claim that under the assumption N +4 where K M depends only on M . It is known from (41) that, for p > N +4 N −4 , v p,α * ≤ C in [0, 1], where C > 0 is independent of p. Therefore, k is the first eigenvalue of ∆ 2 kφ = φ in [0, 1), φ(1) = (∆ kφ )(1) − 4φ (1) = 0.
We then claim that the eigenvalues σ (1) k are simple and analytic in p. We can show that σ (1) k is simple for each k by the comparison principle of ∆ 2 in the radial form (see [26]). We can show that σ (1) k (p) is analytic in p by arguments similar to those in the proof of Lemma 3.3 of [20]. Therefore, a variant of the proof of Lemma 3.4 of [20] and arguments similar to those in the proof of Proposition 3.5 of [20] imply that the following proposition holds. Proposition 4.7. For each k the set of numbers p for which σ (1) k (p) = p is finite (maybe empty). In particular, there exist countably many supercritical exponents {p 1 , . . . , p j , . . .} with p j > N +4 N −4 such that (106) holds if and only if p = p j for all j = 1, 2, . . .. Remark 4.8. Since we do not know the exact profile of u p near the maximum point r p , we do not claim that p j → +∞ as j → +∞. Neither do we claim the set {p j } is nonempty.
Combining Corollary 4.6 and Proposition 4.7, we obtain the following theorem Theorem 4.9. Assume that p is such that p = p j for all j = 1, 2, . . .. Then u p is nondegenerate.