Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities
aa r X i v : . [ m a t h . A P ] M a y Asymptotic behavior of positive solutions to anonlinear biharmonic equation near isolatedsingularities
Hui Yang ∗ Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
Abstract
In this paper, we consider the asymptotic behavior of positive solutions of the bihar-monic equation ∆ u = u p in B \{ } with an isolated singularity, where the punctured ball B \{ } ⊂ R n with n ≥ and nn − < p < n +4 n − . This equation is relevant for the Q -curvature problem in confor-mal geometry. We classify isolated singularities of positive solutions and describethe asymptotic behavior of positive singular solutions without the sign assumptionfor − ∆ u . We also give a new method to prove removable singularity theorem fornonlinear higher order equations. Key words:
Biharmonic equations, isolated singularities, asymptotic behavior, posi-tive singular solutions.
Mathematics Subject Classification (2010): 35J30; 35B40; 35B65
In this paper, we study the asymptotic behavior of positive solutions of the biharmonicequation ∆ u = u p in B \{ } (1.1)with an isolated singularity, where the punctured ball B \{ } ⊂ R n with n ≥ and nn − < p < n +4 n − . Here the unit ball B can be replaced by any bounded domain Ω ⊂ R n containing 0. This equation serves as a basic model of nonlinear fourth-orderequations and is also related to the Q -curvature problem in conformal geometry. Equa-tion (1.1) and related equations arise in several models describing various phenomena ∗ E-mail addresses: [email protected]; [email protected]
1n the applied sciences see, for instance, Gazzola, Grunau and Sweers [13]. For anintroduction to the Q -curvature problem see, for instance, Hang and Yang [20].We first recall that the corresponding second order equation (when n ≥ and nn − < p < n +2 n − ) − ∆ u = u p in B \{ } (1.2)was studied by Gidas-Spruck [15] and Caffarelli-Gidas-Spruck [3]. More specifically,the following classification result is obtained. Theorem A ( [3, 15]) Let n ≥ and u ∈ C ( B \{ } ) be a positive solution of (1.2).Assume nn − < p < n + 2 n − . Then either the singularity at x = 0 is removable, or u is a distribution solution in theentire ball B , and lim | x |→ | x | p − u ( x ) = C , where C = (cid:26) n − p − (cid:18) p − nn − (cid:19)(cid:27) p − . In addition, the asymptotic behavior of positive solutions of (1.2) near an isolated sin-gularity was studied by Lions [25] for < p < nn − , by Aviles [1] for p = nn − , byCaffarelli-Gidas-Spruck [3] and by Korevaar-Mazzeo-Pacard-Schoen [22] in the case p = n +2 n − and by Bidaut-V´eron and V´eron [2] when p > n +2 n − . Hence the isolatedsingularities of positive solutions for the second order equation (1.2) have been verywell understood. The asymptotic behavior of positive solutions for a more generalsecond order equation − ∆ u = K ( x ) u n +2 n − with isolated singularity was studied byChen-Lin [7, 8] and Taliaferro-Zhang [30]. See also Gonz´alez [16], Li [23] and Han-Li-Teixeira [19] for a fully nonlinear equation of second order.In the fundamental paper [24], Lin classified all positive smooth entire solutionsof (1.1) with < p ≤ n +4 n − in R n via the moving plane method. We refer to Chang-Yang [6], Martinazzi [26] and Wei-Xu [31] for the classification of smooth solutionsof the higher-order equations in R n . For the supercritical case, that is for p > n +4 n − ,the positive smooth radial symmetric solutions of (1.1) in R n were studied by Gazzola-Grunau [12], Guo-Wei [17] and Winkler [32]. We also refer to a recent paper Frank-K¨onig [11] for a classification of positive singular solutions to (1.1) with p = n +4 n − in R n \{ } , where the positive singular solutions are radially symmetric about the origin(see Theorem 4.2 in [24]).As far as we know, the classification of isolated singularities of positive solutionsand the asymptotic behavior of positive singular solutions to fourth order equation(1.1) in B \{ } are far less known than the second order problem (1.2). Remark that,a positive solution u of (1.1) in B \{ } may not be radially symmetric.2f one looks closely at the tools being used in the proofs of second order problems,then one finds that the maximum principle plays an essential role. This is a crucialdistinction from higher order problems for which there is no the maximum principle.Here and in the sequel ”higher order” means order at least four. Another important toolintensively used for second order problems is the truncation method. This method ispowerful in regularity theory and in properties of first order Sobolev spaces. However,the truncation method also fails for higher order problems. Therefore, the methodsof above mentioned papers for second order problems cannot be applied to the fourthorder equation (1.1).Nevertheless we succeed here in proving exact asymptotic behavior of positive sin-gular solutions for (1.1) which is completely analogous to its second order counterpart.Remark that our proof is very different from that of Theorem A in [3, 15]. Our mainresult is the following Theorem 1.1.
Let n ≥ and u ∈ C ( B \{ } ) be a nonnegative solution of (1.1) .Assume nn − < p < n + 4 n − . Then either the singularity at x = 0 is removable, or u is a distribution solution in theentire ball B , and lim | x |→ | x | p − u ( x ) = C p,n > , (1.3) where C p,n = [ K ( p, n )] p − and K ( p, n ) = 8( p − h ( n − n − p − + 2( n − n + 20)( p − − n − p −
1) + 32 i . (1.4) Remark 1.1.
We don’t need any additional assumptions for − ∆ u in B \{ } and forboundary conditions. Soranzo [29] studied the local behavior of positive solutions of (1.1) with additional assumption − ∆ u ≥ in B \{ } . (1.5) Under the assumption (1.5) , Soranzo [29] classified the isolated singularities of pos-itive solutions of (1.1) for < p < nn − and obtained an upper bound of radiallysymmetric positive solutions of (1.1) for p ≥ nn − . Theorem 1.1 also answers an openquestion raised in [29] (see Remark 5 there) and shows, in particular, that the nonneg-ativity of − ∆ u in this problem is not necessary. Recently, Jin and Xiong [21] provedsharp blow up rates and the asymptotic radial symmetry of positive solutions of (1.1) with p = n +4 n − near the singularity under the sign assumption (1.5) . We also mentionthat Ferrero-Grunau [10] have obtained the asymptotic behavior of positive radial sin-gular solutions for biharmonic operator and power-like nonlinearity with the Dirichletboundary condition. emark 1.2. In R n , suppose u is a positive smooth function satisfies equation (1.1) with p > , then necessarily we have − ∆ u > in R n . See Theorem 3.1 in Wei-Xu [31]. This important fact about − ∆ u enables the maximumprinciple to be applied to positive solutions of (1.1) in R n . Such as see [24,31]. Hencethe positive solutions of equation (1.1) in R n provide enough information for applyingthe maximum principle, but this is not true for (1.1) in B \{ } . Remark 1.3.
When nn − < p < n +4 n − , it is well known that the function u ( x ) = C p,n | x | − p − is an exact positive singular solution of (1.1) which obviously satisfies asymptotic be-havior (1.3) . See also Guo-Wei-Zhou [18] for another a family of positive singularradial solutions of (1.1) in R n \{ } . The rest of this paper is organized as follows. In Section 2, we establish some basicestimates. In Section 3, we prove Theorem 1.1.
In this section we establish some basic estimates. First we recall the following Liouvilletype theorem. For its proof, such as see Lin [24].
Theorem 2.1. ( [24]) Suppose that u is a nonnegative solution of ∆ u = u p in R n (2.1) for < p < n +4 n − . Then u ≡ in R n . By a doubling lemma of Pol´acik, Quittner and Souplet [28] and above Liouvilletheorem, we have the following singularity and decay estimates. Because their proof issimilar, we only give the proof of decay estimates here.
Lemma 2.1.
Let u ∈ C ( B \{ } ) be a nonnegative solution of (1.1) with < p < n +4 n − . Then u ( x ) ≤ C | x | − p − f or | x | ≤ , (2.2) where C is a constant, depending on n and p only. Remark 2.1.
For the second order equation (1.2) , if one has an upper estimate similarto (2.2) , then one can easily obtain the following Harnack inequality sup r ≤| x |≤ r u ≤ C inf r ≤| x |≤ r u, (2.3) where C is independent of r . Such as see [1,8,15,22]. This is an essential tool for thesepapers to study isolated singularities of second order problems. In a recent paper [4] affarelli, Jin, Sire and Xiong use a similar Harnack inequality to classify isolatedsingularities of positive solutions of a fractional equation. However, this Harnackinequality does not generally hold for fourth order equation (1.1) . In particular, if wesuppose additionally that − ∆ u ≥ in B \{ } , (2.4) then Caristi-Mitidieri [5] proved that the similar Harnack inequality still holds forfourth order equation (1.1) . Remark 2.2.
We also remark that the condition (2.4) is necessary for the validity of theHarnack inequality to biharmonic equations as the following simple example shows:consider the function u ( x ) = P ni =1 x i . It is nonnegative, satisfies ∆ u = 0 and ∆ u = 2 n , but the Harnack inequality does not hold in B (0) . Lemma 2.2.
Let u be a nonnegative solution of ∆ u = u p in B c , (2.5) where B c := { x ∈ R n : | x | > } . Assume < p < n +4 n − . Then u ( x ) ≤ C | x | − p − f or | x | > , (2.6) where C is a constant, depending on n and p only.Proof. Suppose by contradiction that there exist a sequence of nonnegative solutions ( u k ) k of (2.5) and a sequence of points | x k | > , such that M k ( x k ) d ( x k ) > k, k = 1 , , · · · , where M k ( x ) := ( u k ( x )) p − and d ( x ) := dist ( x, ∂B c ) = | x | − for x ∈ B c . By thedoubling lemma of [28] there exists another sequence y k ∈ B c such that M k ( y k ) d ( y k ) > k, M k ( y k ) ≥ M k ( x k ) and M k ( z ) ≤ M k ( y k ) for any | z − y k | ≤ kλ k . where λ k := M k ( y k ) − . We now define ¯ u k ( x ) = λ p − k u k ( y k + λ k x ) for x ∈ B k (0) . Then ¯ u k is a nonnegative solution of ∆ ¯ u k = (¯ u k ) p in B k (0) . Moreover, ¯ u k (0) = 1 and max B k (0) | ¯ u k | ≤ p − . (2.7)By the elliptic estimates, we deduce that a subsequence of (¯ u k ) k converges in C loc ( R n ) to a nonnegative solution u ∞ of (2.1) in R n . By (2.7), we have u ∞ (0) = 1 . Thiscontradicts Theorem 2.1. 5 orollary 2.1. Let u ∈ C ( B \{ } ) be a nonnegative solution of (1.1) with < p < n +4 n − . Then there exists a constant C = C ( n, p ) such that for all | x | ≤ , X k ≤ | x | p − + k |∇ k u ( x ) | ≤ C . (2.8) Proof.
For any x with | x | ≤ , take λ = | x | and define ¯ u ( x ) = λ p − u ( x + λx ) . Then ¯ u is a nonnegative solution of (1.1) in B . By the Lemma 2.1, | ¯ u | ≤ C in B .The standard elliptic estimates give X k ≤ |∇ k ¯ u (0) | ≤ C . Rescaling back we obtain (2.9).Using a similar scaling argument as above, we also have
Corollary 2.2.
Let u be a nonnegative solution of (2.5) with < p < n +4 n − . Then thereexists a constant C = C ( n, p ) such that for all | x | ≥ , X k ≤ | x | p − + k |∇ k u ( x ) | ≤ C . (2.9) In this section we will prove Theorem 1.1. We first show that any nonnegative solutionof (1.1) with p ≥ nn − is a solution in B in the sense of distribution. Lemma 3.1.
Assume p ≥ nn − and that u ∈ C ( B \{ } ) is a nonnegative solution of (1.1) . Then u ∈ L ploc ( B ) and u is a distribution solution of (1.1) in B , that is, Z B u ∆ ϕ = Z B u p ϕ f or all ϕ ∈ C ∞ c ( B ) . (3.1) Proof.
For any < ǫ ≪ , we take η ǫ ∈ C ∞ ( R n ) with values in [0 , satisfying η ǫ ( x ) = ( for | x | ≤ ǫ, for | x | ≥ ǫ (3.2)and |∇ k η ǫ ( x ) | ≤ Cǫ − k for k = 1 , , , . (3.3)Let m = pp − and define ξ ǫ = ( η ǫ ) m . Multiplying (1.1) by ξ ǫ and integrating by partsin B r with < r < , we get Z B r u p ξ ǫ = Z ∂B r ∂∂ν ∆ u + Z B r u ∆ ξ ǫ . | ∆ ξ ǫ | ≤ Cǫ − ( η ǫ ) m − χ { ǫ ≤| x |≤ ǫ } = Cǫ − ( ξ ǫ ) /p χ { ǫ ≤| x |≤ ǫ } . By H¨older’s inequality, we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z B r u ∆ ξ ǫ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ − Z { ǫ ≤| x |≤ ǫ } u ( ξ ǫ ) /p ≤ Cǫ − · ǫ n (1 − /p ) Z { ǫ ≤| x |≤ ǫ } u p ξ ǫ ! /p ≤ C Z { ǫ ≤| x |≤ ǫ } u p ξ ǫ ! /p . Hence we have Z B r u p ξ ǫ ≤ Z ∂B r ∂∂ν ∆ u + C Z { ǫ ≤| x |≤ ǫ } u p ξ ǫ ! /p . This implies that there exists a constant
C > (independent of ǫ ) such that Z B r u p ξ ǫ ≤ C. Now letting ǫ → , we conclude that u ∈ L p ( B r ) .To show that u is a distribution solution we need to establish (3.1). For any ϕ ∈ C ∞ c ( B ) , using η ǫ ϕ as a test function in (1.1) with η ǫ as before gives Z B u ∆ ( η ǫ ϕ ) = Z B u p η ǫ ϕ. (3.4)By a direct computation, we have ∆ ( η ǫ ϕ ) = η ǫ ∆ ϕ + 4 ∇ η ǫ · ∇ ∆ ϕ + 2∆ η ǫ ∆ ϕ + 4 n X i,j =1 ( η ǫ ) x i x j ϕ x i x i + 4 ∇ ∆ η ǫ · ∇ ϕ + ϕ ∆ η ǫ =: η ǫ ∆ ϕ + ψ, and by H¨older’s inequality, we get (cid:12)(cid:12)(cid:12)(cid:12)Z B uψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ − Z { ǫ ≤| x |≤ ǫ } u ≤ Cǫ − · ǫ n (1 − /p ) Z { ǫ ≤| x |≤ ǫ } u p ! /p ≤ C Z { ǫ ≤| x |≤ ǫ } u p ! /p → as ǫ → . ǫ → in (3.4), then (3.1) follows immediately from the dominated convergencetheorem and the proof is complete.Now we prove that if u is a nonnegative solution of (1.1) in R n \{ } , then the signcondition − ∆ u ≥ in R n \{ } holds. This allows us to use the maximum principle for u in R n \{ } . Lemma 3.2.
Assume nn − < p < n +4 n − and that u ∈ C ( R n \{ } ) is a nonnegativesolution of ∆ u = u p in R n \{ } . (3.5) Then − ∆ u is a superharmonic function in R n in the distributional sense. Moreover, − ∆ u ≥ in R n \{ } . Proof.
By Lemma 3.1, we have u ∈ L ploc ( R n ) . Let ϕ ∈ C ∞ c ( R n ) be a nonnegativefunction. We will prove that Z R n ∆ u ∆ ϕ ≥ . Let η ǫ ∈ C ∞ ( R n ) satisfy (3.2) and (3.3). Multiplying (3.5) by η ǫ ϕ and integrating byparts, we obtain ≤ Z R n η ǫ ϕu p = Z R n ∆( η ǫ ϕ )∆ u = Z R n ∆ u (∆ ϕη ǫ + 2 ∇ ϕ · ∇ η ǫ + ϕ ∆ η ǫ ) . Denote ψ = 2 ∇ ϕ · ∇ η ǫ + ϕ ∆ η ǫ . Then ψ ( x ) ≡ for | x | ≤ ǫ and for | x | ≥ ǫ , and | ∆ ψ ( x ) | ≤ Cǫ − . Since n − − np > , we have (cid:12)(cid:12)(cid:12)(cid:12)Z R n ∆ uψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R n u | ∆ ψ |≤ Cǫ − Z { ǫ ≤| x |≤ ǫ } u p ! /p ǫ n (1 − /p ) ≤ Cǫ n − − np → , as ǫ → . Therefore, we obtain Z R n ∆ u ∆ ϕ = lim ǫ → Z R n ∆ u (∆ ϕη ǫ + 2 ∇ ϕ · ∇ η ǫ + ϕ ∆ η ǫ )= Z R n ϕu p ≥ . − ∆ u is a superharmonic function in R n in the distributional sense.Let v ǫ := − ∆ u + ǫ for ǫ > . By Corollary 2.2, we have lim | x |→∞ | ∆ u ( x ) | = 0 .Therefore, for any ǫ > , there exists R ǫ such that v ǫ > ǫ for | x | ≥ R ǫ . Since v ǫ is also a superharmonic function in R n in the distributional sense, we obtain v ǫ ≥ in R n \{ } . Letting ǫ → , we get − ∆ u ≥ in R n \{ } . This completes the proof.Let u be a nonnegative solution of (1.1). We use the following transformation of(1.1) (also known as Emden-Fowler transformation ): set t = ln | x | , θ = x | x | (3.6)and w ( t, θ ) = | x | p − u ( | x | , θ ) = e t/ ( p − u ( e t , θ ) , t ∈ ( −∞ , , θ ∈ S n − . (3.7)By a tedious computation we find that equation (1.1) for u is equivalent to the followingequation for w : ∂ (4) t w + K ∂ (3) t w + K ∂ (2) t w + K ∂ t w + ∆ θ w + 2 ∂ (2) t ∆ θ w + K ∂ t ∆ θ w + J ∆ θ w + K w = w p in ( −∞ , × S n − , (3.8)where ∆ θ is the Beltrami-Laplace operator on S n − , the constants K i = K i ( p, n )( i = 0 , · · · , and J = J ( p, n ) are given by K = 8( p − h ( n − n − p − + 2( n − n + 20)( p − − n − p −
1) + 32 i ,K = − p − h ( n − n − p − + 4( n − n + 20)( p − − n − p −
1) + 128 i ,K = 1( p − h ( n − n + 20)( p − − n − p −
1) + 96 i ,K = 2 p − h ( n − p − − i ,J = − p − h ( n − p − + 4( n − p − − i . Note that if p < n +4 n − , then ( n − p − < . (3.9)9t is not difficult to show that K = K = 0 if p = n + 4 n − . (3.10)Moreover, we have Lemma 3.3.
Assume n ≥ and nn − < p < n +4 n − . Then K > , K > , K < . (3.11) Remark 3.1.
We emphasize that the sign of K and K will be essentially used in ourarguments. We also point that J < for nn − < p < n +4 n − and the sign of K dependson p and n .Proof. By (3.9), we easily obtain K < . Next we will prove that K > under theassumptions. For this purpose, we consider the function f ( s ) = ( n − n − s + 4( n − n + 20) s − n − s + 128 with s ∈ ( n − , n − ) . Then f ′ ( s ) = 3( n − n − s + 8( n − n + 20) s − n − . Since f ′ (0) < , f ′ has only one positive root, we denote it by s + . We also denote s = 4 n − and s = 8 n − . By a direct calculation, we have f ′ ( s ) = n − n +8) n − > . Hence we must have s + < s . We consider separately the case s ≥ s + and the case s < s + . Case 1: s ≥ s + . In this case we have f ′ ( s ) > for all s ∈ ( s , s ) . By (3.10), f ( s ) < f ( s ) = 0 for any s ∈ ( s , s ) . Case 2: s < s + . In this case we have f ′ ( s ) < in ( s , s + ) and f ′ ( s ) > in ( s + , s ) . Combining (3.10) and the basic fact f ( s ) = − n − n − < , we obtain f ( s ) < max { f ( s ) , f ( s ) } = 0 for any s ∈ ( s , s ) . From these we easily get K > if nn − < p < n +4 n − .Now we check K > . Similarly, we consider g ( s ) = ( n − n − s + 2( n − n + 20) s − n − s + 32 with s > n − . Then g ′ ( s ) = 3( n − n − s + 4( n − n + 20) s − n − . Direct calculations show that g ′ ( s ) = n − n − > and g ( s ) = 0 . From this we get g ′ ( s ) > for all s > s and then g ( s ) > g ( s ) = 0 for any s > s . Hence we have K > if p > nn − . 10ext we will establish an important monotonicity formula. Let w be a nonnegativesolution of (3.8). Define E ( t ; w ) := Z S n − ∂ (3) t w∂ t w − Z S n − h (cid:0) ∂ t w (cid:1) − K ∂ t w∂ t w − K ( ∂ t w ) i + 12 Z S n − h | ∆ θ w | − J |∇ θ w | i + K Z S n − w − p + 1 Z S n − w p +1 − Z S n − | ∂ t ∇ θ w | . Then we have the following
Lemma 3.4.
Assume nn − < p < n +4 n − and that w is a nonnegative C solution of (3.8) . Then, E ( r ; w ) is non-increasing in t ∈ ( −∞ , . Furthermore, we have ddt E ( t ; w ) = K Z S n − h(cid:0) ∂ t w (cid:1) + | ∂ t ∇ θ w | i − K Z S n − ( ∂ t w ) . (3.12) Remark 3.2.
An analogous monotonicity formula has been derived by the authorand Zou [33] to study isolated singularities for a fractional equation. Ghergu-Kim-Shahgholian [14] also obtained a similar monotonicity formula for a second ordersemilinear elliptic system with power-law nonlinearity.Proof.
Note that ∂ (4) t w∂ t w = ∂ t (cid:16) ∂ (3) t w∂ t w (cid:17) − ∂ (3) t w∂ t w = ∂ t (cid:16) ∂ (3) t w∂ t w − (cid:0) ∂ t w (cid:1) (cid:17) ,∂ (3) t w∂ t w = ∂ t (cid:0) ∂ t w∂ t w (cid:1) − (cid:0) ∂ t w (cid:1) ,∂ t w∂ t w = 12 ∂ t ( ∂ t w ) . Therefore, multiplying Eq. (3.8) by ∂ t w and integrating by parts on S n − , we get ddt E ( t ; w ) = K Z S n − h(cid:0) ∂ t w (cid:1) + | ∂ t ∇ θ w | i − K Z S n − ( ∂ t w ) . By Lemma 3.3, we have K > and K < . Hence ddt E ( t ; w ) ≤ and we finish theproof. Lemma 3.5.
Let w be a nonnegative C solution of (3.8) with < p < n +4 n − . Then w , ∂ t w , ∂ t w , ∂ (3) t w , ∆ θ w and |∇ θ w | are uniformly bounded in ( −∞ , − ln 2) × S n − .Proof. Define u ( x ) = | x | − p − w ( t, θ ) , t = ln | x | and θ = x | x | . Then u is a nonnegative solution of (1.1). By Lemma2.1, we know that w is uniformly bounded. By Corollary 2.1 we have | ∂ t w | + |∇ θ w | ≤ C X i =0 | x | p − + i |∇ ix u | ≤ C, | ∂ t w | + | ∆ θ w | ≤ C X i =0 | x | p − + i |∇ ix u | ≤ C, | ∂ (3) t w | ≤ C X i =0 | x | p − + i |∇ ix u | ≤ C. Thus the desired conclusion follows.Assume nn − < p < n +4 n − , from Lemmas 3.4 and 3.5 we deduce that the limit lim t →−∞ E ( t ; w ) exists. Let u be a nonnegative solution of (1.1), we define e E ( r ; u ) := E ( t ; w ) , (3.13)where t = ln r and w is defined as in (3.7). Then we have e E (0; u ) := lim r → + e E ( r ; u ) = lim t →−∞ E ( t ; w ) . For any λ > , define u λ ( x ) := λ p − u ( λx ) . Then u λ is also a nonnegative solution of (1.1) in B /λ \{ } . Moreover, we have e E ( r ; u λ ) = E ( t ; w ( · + ln λ, · ))= E ( t + ln λ, w )= E ( λr, u ) . That is, we get the following scaling invariance e E ( r ; u λ ) = e E ( λr ; u ) . (3.14) Lemma 3.6.
Let u ∈ C ( B \{ } ) be a nonnegative solution of (1.1) with nn − < p < n +4 n − . Then either lim | x |→ | x | p − u ( x ) = 0 or lim | x |→ | x | p − u ( x ) = K p − , where K is given by (1.4) . roof. First we compute the possible values of e E (0; u ) . By Lemma 2.1, u λ are uni-formly bounded in C ,α ( K ) on every compact set K ⊂ B / λ \{ } , with some <α < . Therefore, there exists a nonnegative function u ∈ C ( R n \{ } ) , such that upto a subsequence of λ → , u λ converges to u in C loc ( R n \{ } ) . Further, u satisfies ∆ u = u p in R n \{ } . By Lemma 3.2, we have − ∆ u ≥ in R n \{ } . The maximum principle gives thateither u ≡ in R n \{ } or u > in R n \{ } . Therefore, by Theorem 4.2 in [24], u is radially symmetric with respect to the origin0. Moreover, by the scaling invariance of e E , we have for any r > that e E ( r ; u ) = lim λ → e E ( r ; u λ ) = lim λ → e E ( λr ; u ) = e E (0; u ) . (3.15)Let w ( t ) := | x | p − u ( | x | ) , t = ln | x | . Then w satisfies d dt w + K d dt w + K d dt w + K ddt w + K w = ( w ) p in R . (3.16)From (3.15), E ( t ; w ) = e E ( r ; u ) is a constant. By Lemma 3.4, ddt E ( t ; w ) = | S n − | " K (cid:18) d dt w (cid:19) − K (cid:18) ddt w (cid:19) ≡ . Since K < and K > , we get that ddt w ≡ in R and then w is a constant. By(3.16), either w = 0 or w = K p − . Hence, by (3.15) we obtain e E (0; u ) ∈ (cid:26) , (cid:18) − p + 1 (cid:19) K p +1 p − | S n − | (cid:27) . If e E (0; u ) = 0 , then u ≡ . Since this function u is unique, we conclude that u λ → for any sequence of λ → , in C ( K ) on every compact set K ⊂ R n \{ } .Therefore, we easily get lim | x |→ | x | p − u ( x ) = 0 . If e E (0; u ) = (cid:16) − p +1 (cid:17) K p +1 p − | S n − | , then we have u ( x ) ≡ K p − | x | − p − .
13n this case the function u is also unique, so we obtain that u λ → K p − | x | − p − forany sequence of λ → , in C ( K ) on every compact set K ⊂ R n \{ } . In particular,we have u λ ( x ) → K p − as λ → in C ( S n − ) . We quickly get lim | x |→ | x | p − u ( x ) = K p − . This completes the proof.
Lemma 3.7.
Assume nn − < p < n +4 n − and that u ∈ C ( B \{ } ) is a nonnegativesolution of (1.1) . If lim | x |→ | x | p − u ( x ) = 0 , (3.17) then Z {| x |≤ / } u ( p − n/ < + ∞ . (3.18) Proof.
Set ϕ ( x ) = | x | − n − p − ( p − nn − ) . We recall that in radial coordinates r = | x | , we have ∆ ϕ ( r ) = ϕ (4) ( r ) + 2( n − r ϕ (3) ( r ) + ( n − n − r ϕ ′′ ( r ) − ( n − n − r ϕ ′ ( r ) (3.19)Denote γ := − n − p − (cid:18) p − nn − (cid:19) < . Direct calculations show that ∆ ϕ = h γ ( γ − γ − γ −
3) + 2 γ ( n − γ − γ − γ ( n − n − γ − − γ ( n − n − i r γ − = γ ( γ − h ( γ − γ −
3) + 2( n − γ −
1) + ( n − n − i r γ − = γ ( γ − h ( γ − γ + n −
4) + ( n − γ + n − i r γ − = γ ( γ − γ + n − γ + n − r γ − . Since γ + n − p − > , we have A := γ ( γ − γ + n − γ + n − > . ∆ ϕϕ = A | x | in R n \{ } . (3.20)For small ǫ > , let ζ ǫ be a smooth cut-off function satisfying ζ ǫ ( x ) = ( for ǫ ≤ | x | ≤ , for | x | ≤ ǫ , | x | ≥ (3.21)and |∇ k ζ ǫ ( x ) | ≤ Cǫ − k for k = 1 , , , . (3.22)Using ζ ǫ ϕ as a test function in (1.1) and integrating by parts we obtain Z B ζ ǫ uϕ (cid:18) ∆ ϕϕ − u p − (cid:19) = − Z B uF ( ζ ǫ , ϕ ) , (3.23)where F ( ζ ǫ , ϕ ) =4 ∇ ζ ǫ · ∇ ∆ ϕ + 2∆ ζ ǫ ∆ ϕ + 4 n X i,j =1 ( ζ ǫ ) x i x j ϕ x i x i + 4 ∇ ∆ ζ ǫ · ∇ ϕ + ϕ ∆ ζ ǫ . By (3.21), (3.22) and Lemma 2.1, we estimate (cid:12)(cid:12)(cid:12)(cid:12)Z B uF ( ζ ǫ , ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z { ≤| x |≤ } uF ( ζ ǫ , ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z { ǫ ≤| x |≤ ǫ } uF ( ζ ǫ , ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C Z { ǫ ≤| x |≤ ǫ } u (cid:20) ǫ | x | γ − + 1 ǫ | x | γ − + 1 ǫ | x | γ − + 1 ǫ | x | γ (cid:21) ≤ C + C ǫ n ǫ γ − ǫ − p − ≤ C + C < + ∞ , where C = C ( p, n, u ) and C = C ( p, n ) are two positive constants (independentof ǫ ). Hence Z B ζ ǫ uϕ (cid:18) ∆ ϕϕ − u p − (cid:19) ≤ C < + ∞ (3.24)uniformly in ǫ . By the assumption (3.17), u p − ( x ) = o (1) | x | − as | x | → . This together with (3.20) and (3.24) gives Z B ζ ǫ u | x | γ − ≤ C < + ∞ , C is a positive constant independent of ǫ . Therefore, by Lemma 2.1, Z { ǫ ≤| x |≤ } u ( p − n/ = Z { ǫ ≤| x |≤ } uu (( p − n − / ≤ C ( p, n ) Z { ǫ ≤| x |≤ } u | x | − (( p − n − / ( p − = C ( p, n ) Z { ǫ ≤| x |≤ } u | x | γ − ≤ C ( p, n ) Z B ζ ǫ u | x | γ − ≤ C ( p, n ) C < + ∞ . Letting ǫ → , we get (3.18) by the dominated convergence theorem.Now we give a new method to obtain the removable singularity theorem. For ourfourth order equation (1.1), the classical methods based on the maximum principle tosecond order problems (such as see [1, 4, 8, 22]) fail. We remark that our method alsoapply to higher order equations. This method is based on the following RegularityLifting Theorem from Chen-Li [9].Let V be a Hausdorff topological vector space. Suppose there are two extendednorms (i.e., the norm of an element in V might be infinity) defined on V , k · k X , k · k Y : V → [0 , + ∞ ] . Let X := { v ∈ V : k v k X < + ∞} and Y := { v ∈ V : k v k Y < + ∞} . Assume that spaces X and Y are complete under the corresponding norms and theconvergence in X or in Y implies the convergence in V . Theorem 3.1. ( [9], Theorem 3.3.1) Let T be a contraction map from X into itself andfrom Y into itself. Assume that f ∈ X and that there exists a function g ∈ Z := X ∩ Y such that f = T f + g in X . Then f also belongs to Z . Remark 3.3.
We usually choose V to be the space of distributions, and X and Y to befunction spaces, for instance, X = L p and Y = W ,q . Next we will use this Regularity Lifting Theorem to prove a removable singularityresult.
Lemma 3.8.
Assume nn − < p < n +4 n − and that u ∈ C ( B \{ } ) is a nonnegativesolution of (1.1) . If Z {| x |≤ / } u ( p − n/ < + ∞ , (3.25) then the singularity at x = 0 is removable, i.e., u ( x ) can be extended to a C solutionof (1.1) in the entire ball B . emark 3.4. For the second order equation (1.2) , a similar result for removable sin-gularity was proved by Gidas-Spruck [15]. However, their proof is based on a doubleapplication of the De Giorgi-Nash-Moser bootstrap arguments, which cannot be ap-plied to our fourth order problem (1.1) .Proof.
Let G ( x, y ) be the Green’s function of ∆ in B / with homogeneous Dirich-let boundary conditions, Then, for each fixed y ∈ B , G ( · , y ) is a distributional solu-tion of ( ∆ G ( · , y ) = δ ( · − y ) in B / ,G ( · , y ) = ∂G ( · ,y ) ∂ν = 0 on ∂B / , and there exists positive constant C n such that < G ( x, y ) ≤ Γ ( | x − y | ) := C n | x − y | − n for x, y ∈ B / , x = y. Define v ( x ) := − u ( x ) + Z B / G ( x, y ) u p ( y ) dy, x ∈ B / , then v ∈ L ( B / ) . Moreover, by Lemma 3.1, v satisfies ∆ v = 0 in B / in the distributional sense. Using Theorem 7.23 in [27], we get v ∈ L ∞ loc ( B / ) .Now we split the right hand side of (1.1) into two parts: u p = u p − u := a ( x ) u. Then, by the assumption (3.25), a ( x ) ∈ L n ( B / ) . For any positive number L > ,let a L ( x ) = ( a ( x ) if | a ( x ) | ≥ L, otherwise , and a M ( x ) = a ( x ) − a L ( x ) . Define the linear operator ( T L w )( x ) = Z B / G ( x, y ) a L ( y ) w ( y ) dy. Then u satisfies the equation u ( x ) = ( T L u )( x ) + F L ( x ) in B / , (3.26)where F L ( x ) = Z B / G ( x, y ) a M ( y ) u ( y ) dy − v ( x ) + h ( x ) h ( x ) = Z { ≤| y |≤ } G ( x, y ) u p ( y ) dy. Note that | h ( x ) | ≤ C Z { ≤| y |≤ } G ( x, y ) dy ≤ C Z { ≤| y |≤ } | x − y | − n dy ≤ C Z B | y | − n dy ≤ C for all x ∈ B / . Hence v, h ∈ L ∞ ( B / ) .We will prove that, for any nn − < q < ∞ ,(1) T L is a contracting operator from L q ( B / ) to L q ( B / ) for L large.(2) F L ∈ L q ( B / ) .Then, by the Regularity Lifting Theorem 3.1, we obtain u ∈ L q ( B / ) for any nn − < q < ∞ .(1) The estimate of the operator T L . For any nn − < q < ∞ , there exists < r < n such that q = 1 r − n . By Hardy-Littlewood-Sobolev inequality and H¨older inequality, we have k T L w k L q ( B / ) ≤ k Γ ∗ a L w k L q ( R n ) ≤ C k a L w k L r ( B / ) ≤ k a L k L n ( B / ) k w k L q ( B / ) . Since a ( x ) ∈ L n ( B / ) , we can choose L sufficiently large, such that k a L k L n ( B / ) ≤ . Therefore, T L : L q ( B / ) → L q ( B / ) is a contracting operator for L large.(2) The integrability of the function F L ( x ) . Obviously, we only need to show that, for any nn − < q < ∞ , F L ( x ) := Z B / G ( x, y ) a M ( y ) u ( y ) dy ∈ L q ( B / ) . Since a M ( x ) is a bounded function, we have k F L k L q ( B / ) ≤ k a M u k L r ( B / ) ≤ C k u k L r ( B / ) .
18y the assumption (3.25), u ∈ L r ( B / ) for any < r ≤ ( p − n . Note that q = ( p − n − p ) if r = ( p − n . Hence, we conclude that, for the following values of q , ( < q < ∞ if p ≥ , < q ≤ ( p − n − p ) if p < ,F L ( x ) ∈ L q ( B / ) .Using the Regularity Lifting Theorem 3.1, we obtain ( u ∈ L q ( B / ) for any < q < ∞ if p ≥ ,u ∈ L q ( B / ) for any < q ≤ ( p − n − p ) if p < . Now we note that from the starting point where u ∈ L ( p − n ( B / ) , we get u ∈ L r ( B / ) with r = ( p − n − p ) , p < . By a similar argument as above, we get ( u ∈ L q ( B / ) for any < q < ∞ if p ≥ ,u ∈ L q ( B / ) for any < q ≤ ( p − n − p ) if p < . Hence by iteration we have for k = 1 , , · · · , ( u ∈ L q ( B / ) for any < q < ∞ if p ≥ k +1 k ,u ∈ L q ( B / ) for any < q ≤ ( p − n k +1) − kp ] if p < k +1 k . This implies that for any fixed dimension n , a finite number of iterations gives u ∈ L q ( B / ) for any < q < ∞ . Finally, By H¨older inequality, we have Z B / G ( x, y ) u p ( y ) dy ∈ L ∞ ( B / ) , From this and (3.26) we easily deduce that u ∈ L ∞ ( B / ) . By estimates of ellipticequations, u ( x ) is smooth at 0. Therefore 0 is a removable singularity. Proof of Theorem 1.1.
The proof of Theorem 1.1 is now just a combination of Lemmas3.6, 3.7 and 3.8. (cid:3) cknowledgments. The author would like to thank Professor Sun-Yung A. Changfor many helpful discussions and comments. The author would also like to thank hisadvisor Professor Wenming Zou for his constant support and encouragement. Thiswork was done during the author’s visit to Princeton University. He thanks TsinghuaUniversity for funding his visit and thanks the Department of Mathematics at PrincetonUniversity for kind hospitality.
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