Radial symmetry of positive entire solutions of a fourth order elliptic equation with a singular nonlinearity

The necessary and sufficient conditions for a regular positive entire solution $u$ of the biharmonic equation: \begin{equation} \label{0.1} -\Delta^2 u=u^{-p} \;\; \mbox{in $\R^N \; (N \geq 3)$}, \;\; p>1 \end{equation} to be a radially symmetric solution are obtained via the moving plane method (MPM) of a system of equations. It is well-known that for any $a>0$, \eqref{0.1} admits a unique minimal positive entire radial solution ${\underline u}_a (r)$ and a family of non-minimal positive entire radial solutions $u_a (r)$ such that $u_a (0)={\underline u}_a (0)=a$ and $u_a (r) \geq {\underline u}_a (r)$ for $r \in (0, \infty)$. Moreover, the asymptotic behaviors of ${\underline u}_a (r)$ and $u_a (r)$ at $r=\infty$ are also known. We will see in this paper that the asymptotic behaviors similar to those of ${\underline u}_a (r)$ and $u_a (r)$ at $r=\infty$ can determine the radial symmetry of a general regular positive entire solution $u$ of \eqref{0.1}. The precisely asymptotic behaviors of $u (x)$ and $-\Delta u (x)$ at $|x|=\infty$ need to be established such that the moving-plane procedure can be started. We provide the necessary and sufficient conditions not only for a regular positive entire solution $u$ of \eqref{0.1} to be the minimal entire radial solution, but also for $u$ to be a non-minimal entire radial solution.


Introduction
We consider radial symmetry of positive entire solutions of the equation where N = 3, 1 < p < 3 and N ≥ 4, p > 1. The necessary and sufficient conditions for a positive entire solution of (1.1) to be a positive entire radially symmetric solution are established. Equation (1.1) has been extensively studied in recent years, see, for example, [1,2,3,5,9,13,14,17,19,20,21] and the references therein. It arises in the study of the deflection of charged plates in electrostatic actuators in the modeling of electrostatic micro-electromechanical systems (MEMS) (see [18,22] and the references therein).
It is known from [5] that for N = 3 and 1 < p < 3; N ≥ 4 and p > 1 (1.1) admits a singular entire radial solution: U s (r) = Lr α , r = |x|, where and in the following, Moreover, for any a > 0, there is a uniqueb := b(a) > 0 such that the problem It is also known from [5] that for any b <b, (1.3) does not admit an entire radial solution; for any b >b, (1.3) admits a unique entire radial solution u a,b (r) which has the growth rate O(r 2 ) at r = ∞. Therefore we see that the behaviors of the minimal and non-minimal entire solutions at ∞ are different. A comparison principle (Lemma 3.2 in [20]) ensures that u a,b > u a,b in (0, ∞) for b >b. These imply that for any a > 0, u a,b is the (unique) minimal positive entire radial solution of (1.1) and {u a,b } b>b are a family of entire non-minimal radial solutions of (1.1). Meanwhile, the comparison principle also implies that for any b 1 > b 2 >b, u a,b 1 > u a,b 2 in (0, ∞). The stability of positive entire solutions of (1.1) has also been studied in [14] and the references therein.
In this paper, we are interested in the relationship between the radial symmetry and the asymptotic behavior at ∞ of a positive entire solution of (1.1). We will see that if a positive regular entire solution u of (1.1) admits the asymptotic behavior as that of the minimal entire radial solution of (1.1), it is actually the minimal entire radial solution of (1.1) with respect to some x * ∈ R N . Meanwhile, if a positive regular entire solution u of (1.1) admits the asymptotic behavior as that of a nonminimal entire radial solution of (1.1), it is actually a non-minimal entire radial solution of (1.1) with respect to some x * ∈ R N .
Our main results are the following theorems. Then u is the minimal radial entire solution of (1.1) with the initial value u(x * ) at some x * ∈ R N (i.e. u(x) = u(r) with r = |x − x * |) if and only if for N = 4, ∞, for N = 5.
Note that 5 − N − 2α ∈ (−1, 0) when N = 3, 4 or 5 and p ∈ ( N +2 6−N , p * ). The following theorem provides the necessary and sufficient conditions for a positive entire solution of (1.1) to be a non-minimal positive radial entire solution of (1.1). Theorem 1.4. Let u ∈ C 4 (R N ) be a positive entire solution of (1.1) with N = 3 and 1 < p < 3; N ≥ 4 and p > 1. Then u is an entire radial solution about some x * ∈ R N , but is not the minimal positive entire radial solution about x * of (1.1), if and only if there exists D > 0 such that The constant D then determines a particular non-minimal positive entire radial solution.
Theorems 1. 1-1.4 show that the asymptotic behavior given in (1.6), (1.8), (1.9) or (1.10) near ∞ of a positive entire solution u of (1.1) determines its radial symmetry 3 with respect to some x * ∈ R N , which seem to be the first such kinds of results for problem (1.1).
Let us comment on some related results. The semilinear equations (P ) −∆u = u p in R N (N ≥ 3), p > N + 2 N − 2 and (Q) ∆u = u −p in R N (N ≥ 2), p > 0 have been studied in the past few decades. Some sufficient conditions for a regular positive entire solution of (P) and (Q) to be an entire radial solution are given in [25] for (P) provided p ∈ ( N +2 N −2 , N +1 N −3 ) and in [15] for (Q) provided p > 0 respectively. The results in [25] were generalized to p ≥ N N −4 for N ≥ 5 in [10]. Recently, the necessary and sufficient conditions for an entire solution u of the equation to be the entire radial solution of (P 1 ) with the initial value at some x * ∈ R N are provided in [11]. Note that (1.1) can be written to the following system of equations: As in [11], we use the moving plane method for a system of equations to obtain our results, but we need to do more delicate estimates for the solution u and ∆u near ∞, since (Q) has a more complicated structure of solutions than (P1). We discuss not only the minimal solution but also the non-minimal solutions in this paper. Such estimates we need to do are more complicated since they rely on two parameters p and N. Moreover, for the non-minimal entire radial solution case, the asymptotic behavior (1.10) is not enough to make the moving-plane procedure works, we need to obtain more detailed information of the asymptotic behavior of u based on (1.10).
To know more information of the positive entire solutions with asymptotic behavior (1.6) near ∞, we use a Kelvin type transformation: and make a fundamental estimate for The key point is to show that W (s) is Lipschitz continuous, or Hölder continuous in some case, in a neighborhood of s = 0.
In Sections 2-5, we deal with positive entire solutions u of (1.1) with the asymptotic behavior (1.6). In the last section, we deal with positive entire solutions u of (1.1) with (1.10). In Section 2, we first introduce some preliminary results about the eigenvalues and eigenfunctions of ∆ 2 S N−1 . Then, using the Kelvin-type transformation given in (1.12) we obtain the information of v(y) near y = 0. In Section 3, we derive an important estimate for W (s) (given in (1.13)) near s = 0. In Section 4, some estimates for v(s) and v(s, θ) near s = 0 are obtained. We present the proofs of Theorems 1.1, 1.2 and 1.3 in Section 5. Finally, we prove Theorem 1.4 in Section 6. In this paper, we use C to denote a positive constant which may change line by line.

Preliminaries
In this section, we present some results which will be useful in the following proofs. We use the spherical coordinates x = (r, θ) as usual. First, let us to show the following lemma (see Lemma 2.1 in [11]). Lemma 2.1. Let (λ, Q(θ)) be a pair of eigenvalue and eigenfunction of the equation Then (λ 2 , Q(θ)) is a pair of eigenvalue and eigenfunction of the equation Conversely, if (σ, Q(θ)) is a pair of eigenvalue and eigenfunction of (2.2) with σ = 0, then σ > 0 and (σ 1/2 , Q(θ)) is a pair of eigenvalue and eigenfunction of (2.1).
It is known from [4] that for N ≥ 3, the eigenvalues of the equation (2.1) are given by Then Lemma 2.1 implies that the eigenvalues of the equation (2.2) are λ 2 k with the same multiplicity. In particular, we have The boot-strap argument implies that for 1 ≤ j ≤ m k , with C > 0 being independent of k and τ ≥ 1, τ 1 ≥ 1 being positive integers such that 2τ > N − 1, 2τ 1 > N. In Sections 2-5, we assume that u ∈ C 4 (R N ) is a positive entire solution of (1.1) with (1.6). Introducing the Kelvin-type transformation: we see that u(x) = u(r, θ), v(y) = v(s, θ) with s = |y| = r −1 and with the notations ∂ r = ∂ ∂r and ∂ m r = ∂ m ∂r m for 2 ≤ m ≤ 4. Direct calculations imply that Therefore, the study of the behavior of u near |x| = ∞ is converted to the study of the behavior of v of the equation (2.8) near |y| = 0.
Lemma 2.2. Let u ∈ C 4 (R N ) be a positive entire solution of (1.1) and let v be given in (2.7). Suppose that Then for any integer ℓ ≥ 0 there exist constants M = M(u) > 0, s * = s * (u) > 0 such that Proof. The estimates in (2.10) follow from (2.9) by standard elliptic theory.
Let v be a solution of (2.8).
Then v and w satisfy and ξ(s, θ) is between v(s, θ) and v(s).
To end this section, we notice that since w(s, ·) ∈ H 2 (S N −1 ) ⊂ L 2 (S N −1 ) and w = 0,  In this section, we establish some fundamental estimates of W (s) for s near 0, where W (s) is defined by We will see that the Lipschitz continuity of W (s) at the origin is crucial in proving the expansion of u near ∞, which can be used to obtain the symmetry of u by the moving-plane method. Csβ, for p ∈ ( N +2 6−N , p * ) and N = 3, 4, 5, where p * is given by (1.7).
We now determine the sign of β (k) 3 for any k ≥ 2.
We continue the proof of Proposition 3.1. For any k ≥ 2, from the equation satisfied by z k j and the ODE theory, we see that, for any T > − ln s * * , there exist constants A k j,i , B k i (i = 1, 2, 3, 4) such that for t > T , i . More precisely, the detailed calculations show that Note that Note also that for ℓ = 1, 3 and any fixed t > T , It follows from (3.22) and arguments similar to those in [11] that for t ∈ (T, ∞). This implies that our claim (3.20) holds for z k j (t) ≡ 0. Therefore, our claim (3.20) holds.

Proofs of Theorems 1.1-1.3
In this section, we present the proofs of Theorems 1.1-1.3 by using the well known moving plane method.
For γ ∈ R, define the hyperplane: For any x ∈ R N , denote the reflection point of x about Υ γ by x γ , i.e.
We have the following lemma by using Theorem 4.4.
Assume w(x) = −∆u(x) and rewrite (1.1) in the following form: Let us recall Lemma 4.2 in [23] due to Troy. We obtain readily that Lemma 5.2. Let γ ∈ R and u be a positive entire solution of (1.1). Suppose that Then where x γ is the reflection point of x with respect to Υ γ .
As a consequence of Lemma 5.2, we have the following result.
Proofs of Theorems 1.1, 1.2 and 1. 3 We first show the sufficiency of these theorems. The main idea of the proof is similar to those in [11,25]. We claim that there exists γ ′ > 0 such that Suppose for contradiction that (5.7) does not hold. Then by Lemma 5.3, there exist two sequences {γ j } → ∞ and {x j } with x j < γ j such that Thanks to y j tends to ∞, we see that u(y j ) tends to infinity. In turn |x j | → ∞. By Lemma 5.1, we must have Thus, it follows that, for any γ 1 > γ 0 + 1, since (x j ) γ j 1 ≫ (x j ) γ 1 1 for j large and u(x) → ∞ as |x| → ∞. On the other hand, using Lemma 5.1 again, we conclude that This is a contradiction and (5.7) follows. The rest of the proof is same as that of Theorem 1.1 in [11] and [25] for the sufficiency of Theorems 1.1, 1.2 and 1.3. We omit them here.
when N = 3. We obtain our desired results by using (4.4) and Claim 5 again.
We continue to show the necessity of Theorems 1.
where ǫ 0 is given in Theorem 1.2. It follows from (5.11) that for r sufficiently large, In this section, we present the proof of Theorem 1.4. To do this, we first obtain the asymptotic behavior of a non-minimal positive radial entire solution of (1.1).
We know from [5] that when N = 3 and 1 < p < 3; N ≥ 4 and p > 1, for any 34 fixed a > 0 and ∞ > b >b, (1.3) admits a unique non-minimal positive radial entire solution u a,b (r) such that r −2 u a,b (r) ∈ (A 1 , A 2 ) for r sufficiently large, The following proposition presents the asymptotic behavior of u a,b (r) at r = ∞. Proposition 6.1. There exists d > 0 (d depends on a and b) such that, for r near +∞, Proof. We first show It is easily seen from the equation in (1.3) that ∆u a,b (r) is decreasing in (0, ∞). Therefore, there are three cases for ∆u a,b (r): (i) ∆u a,b (r) → −e < 0 (e may be +∞) as r → ∞, (ii) ∆u a,b (r) → 0 as r → ∞, (iii) ∆u a,b (r) → d > 0 as r → ∞. We show that the cases (i) and (ii) do not happen. Since r −2 u a,b (r) ∈ (A 1 , A 2 ) for r sufficiently large, we have that If (i) occurs, we see that for any small ǫ > 0, there is an R = R(ǫ) > 1 such that (6.5) ∆u a,b (r) < −e + ǫ for r > R.
(We may assume 0 < e < ∞. If e = ∞, we can choose any 0 < e 1 < ∞ such that (6.5) holds.) This implies This implies by sending ǫ to 0. This contradicts to (6.4). If (ii) occurs, arguments similar to those in the proof of case (i) imply that This also contradicts to (6.4). Therefore, case (iii) occurs. Clearly using the arguments similar to those in the proof of case (i), we can prove that and then the limits in (6.3) hold.
Making the transformations: we have that, for t near ∞, w(t) satisfies the equation: The ODE theory implies that for T ≫ 1 sufficiently large and t > T , Note that B 1 and B 2 are independent of T . Since w(t) → 0 as t → ∞, we have that M 1 = 0 and we easily see that This implies that the identities in (6.1) hold. To see the identities in (6.2), we define ̺(r) = r −2 u(r) − d 2N . Then ̺(r) → 0 as r → ∞ and ̺(r) satisfies the equation Making the transformations: we have that, for t near ∞, z(t) satisfies the equation if p = N 2 . Arguments similar to those in the proof of (6.1) imply that for t near +∞, 2 and N = 4, where κ = min{2, N − 2, 2(p − 1)}. This implies that the identities in (6.2) hold. Since u(r) = r 2 ̺(r) + d 2N r 2 , we have that ∆(r 2 ̺(r)) = ∆u(r) − d > 0 for r ∈ (0, ∞).
The proof of this proposition is completed. We also know that lim b→∞ d(a, b) = ∞.
Proof of Theorem 1.4. Without loss of generality, we assume x * = 0 in Theorem 1.4. The necessity follows from Proposition 6.1.
To prove the sufficiency of Theorem 1.4, we need to know more information on the asymptotic behavior of an entire solution u ∈ C 4 (R N ) of (1.1) satisfying (1.10). The main idea is similar to that of the proof of the sufficiency of Theorem 1.1.
Since w(s) = 0, we have the expansion: . .} is given in Section 2. We also see that w i j (s) with 1 ≤ j ≤ m i satisfies the equation (6.11) where λ i = i(N +i−2), i = 0, 1, 2, . . . are the eigenvalues of the equation −∆ S N−1 Q = λQ given by (2.3) and We see that Similar to Proposition 3.1, we have the following result.
In conclusion, we have the following theorem.
Theorem 6.6. Let v be a solution of (6.7) andw be given in (6.32). Then we have (i) v(y) = v(s) + sw(s, θ) satisfies We obtain from Theorem 6.6 the asymptotic expansion of u(x) near |x| = ∞. Proof. To prove (6.47), without loss of generality, we assume that lim j→∞ x j |x j | = θ ∈ S N −1 .
It follows that there exists a sequence of bounded positive numbers {d j } such that u(x j ) > u(x d j ), x d j = x j + (2d j , 0, . . . , 0), ∀j ∈ N.
If the first case occurs, we choose a convergent subsequence of {γ j } (still denoted by {γ j }) with the limit γ ≥ γ 0 + 1 and apply (6.47) and (6.48) to obtain This contradicts (6.50). We can derive a contradiction for the second case similarly. The proof is a little variant of the proof of Lemma 8.2 of [25]. Thus, neither the first nor the second case can occur and (6.49) holds. This completes the proof of this lemma.
To complete the proof of the sufficiency, we use moving-plane arguments of the system of equations (6.46). The proof is exactly the same as the proof of Theorem 1.1. We omit the details here. Remark 6.9. We conjecture that the following conclusion holds: If u ∈ C 4 (R N ) is an entire solution of (1.1) with N = 3 and 1 < p < 3 or N ≥ 4 and p > 1, then u is the minimal radial entire solution of (1.1) about some x * ∈ R N , if and only if (6.51) |x| −2 u(x) → 0 as |x| → ∞.
This conjecture implies that if u is an entire solution of (1.1) and (6.51) holds for u, then u must have the exact asymptotic behavior at ∞: where α and L are given in (1.2).