Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Transcription

1 Liouville-type theorems decay estimates for solutions to higher order elliptic equations Guozhen Lu, Peiyong Wang Jiuyi Zhu Abstract. Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumption of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations systems. Using a rescaling technique doubling lemma developed recently in [PQS], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouvilletype theorems in the recent literature. Moreover, we also investigate the singularity decay estimates of higher order elliptic equations. 1. Introduction The aim of this paper is to establish some Liouville-type theorems without the boundedness assumption on nonnegative solutions to certain classes of elliptic equations. Liouvilletype theorems are powerful tools to prove a priori bounds for nonnegative solutions in a bounded domain. Using the rescaling method (also called the blow-up method) in the elegant paper [GS2], an equation in a bounded domain will blow up to become another equation in the whole Euclidean space or a half space. With the aid of the corresponding Liouville-type theorem in the Euclidean space R N half space R N + a contradiction argument, the a priori bounds could be deduced. Moreover, the existence of nonnegative solutions to elliptic equations is established by the topological degree method using a priori estimates (see e.g. [DLN]). In [PQS], the authors develop a general method for the derivation of universal, pointwise a priori estimates of local solutions from the Liouville-type theorem. Their results show that the universal bounds theorems for local solutions the Liouville-type theorem are essentially equivalent. By further exploring their rescaling method, we also obtain that the boundedness assumption of solutions in proving the Lioville-type theorem is not essential in many cases Mathematics Subject Classification. 35B53, 35J40, 35J47, 35B45, Key words phrases. Higher order elliptic equations, Polyharmonic operators on half spaces, Dirichlet problem, Navier boundary condition, Liouville-type theorem, Decay estimates for solutions, Doubling property, Without boundedness assumptions. Research is partly supported by a US NSF grant #DMS Corresponding Author: Jiuyi Zhu at jiuyi.zhu@wayne.edu. 1

2 2 GUOZHEN LU, PEIYONG WANG AND JIUYI ZHU We denote the Sobolev critical exponent by N+2m if N > 2m, N 2m p S := if N 2m, where m N N is the dimension of Euclidean space R N or half space R N +. The Liouville-type theorem for the subcritical higher order elliptic equations in Euclidean space was established in [WX] as follows, Theorem A: Let 1 < q < p S. If u is a classical nonnegative solution of (1.1) ( ) m u = u q in R N, then u 0. For the higher order elliptic equation in half space, generally speaing, two types of boundary conditions, i.e. the Dirichlet boundary problem the Navier boundary problem, are considered. The Liouville theorem for the Dirichlet boundary type of higher order equation in half space has been obtained in [RW]. Theorem B: Let1 < q < p S. If u is a classical solution of ( ) m u = u q in R N +, (1.2) u 0 in R N +, u = u x N = = u x N = 0 on R N +, then u 0. The Liouville-type theorem for the Navier boundary problem of higher order elliptic equation in half space has been studied in [BM], [GL] [S]. The authors consider higher order elliptic equation (1.3) ( ) m u = u p u 0 in R N +, in R N +, u = u = = u = 0 on R N +. Under the following boundedness assumption on the solution: (1.4) sup u <, sup i u <, i = 1, 2,, m 1. They established that the solutions in (1.3) are trivial if 2m+1 < N 1 < p < N 1+2m N 1 2m or if 2m + 1 N 1 < p <. The proof of their results is based on an idea of [D]. The idea is the following: if there exists a solution of (1.3) one is able to show that any solution is increasing in the x N direction, then passing the limit as x N, one could get a solution of the same equation in R N 1, which in turn allows the use of the Liouville-type theorem in the whole space. (Note that the critical exponent in (1.3) is N 1+2m). We would lie to mention that the Liouville-type theorem for (1.3) in [D] is for N 1 2m the case of m = 1 with the assumption that the solution is bounded. Later on this result was improved to hold for unbounded solution in [BCN], among many other results. We also mention that Liouville-type theorems for differential inequalities systems involving polyharmonic operators related to the Hardy-Littlewood-Sobolev inequalities have also been extensively studied. We refer the reader to the paper by Caristi, D Ambrosio Mitiditieri [CDM] references therein (see also [M1]). A natural question is then whether the boundedness assumption for the Liouvilletype theorem is necessary for higher order elliptic equation or not. In particular, we are

3 LIOUVILLE THEOREMS AND HIGHER ORDER ELLIPTIC EQUATIONS 3 interested in the Liouville-type theorems for solutions to polyharmonic equations. We show that it is indeed unnecessary for such equations establish that Theorem 1. If u is a classical solution of (1.3) with 2m + 1 N 1 < p < N+2m N 2m or with 2m + 1 > N 1 < p <, then u 0. In addition, we consider the a priori estimates of possible singularities for local solutions of higher order elliptic equation (1.5) ( ) m u = f(u) when f satisfies certain conditions. More precisely, we will obtain Theorem 2. Let 1 < p < p S Ω R N be a domain in R N. Assume that the function f : [0, ) R is continuous, (1.6) lim u u p f(u) = ρ (0, ). Then there exists C(N, f, m) > 0 independent of Ω u such that for any positive solution u of (1.5) in Ω, there holds (1.7) 2 D ν u(x) C(N, f, m)(1 + dist 2m (x, Ω)), x Ω, where ν = {ν 1,, ν N }, D ν u = D ν 1 x 1 u D ν N xn u dist(x, Ω) is the distance of x from Ω. In particular, if Ω = B R \ {0} for some R > 0, then 2 D ν u(x) C(N, f, m)(1 + x 2m ), where B R is a ball centered at 0 with radius R. 0 < x R 2, In the special case of f(u) = u p. More precise results can be given in bounded or exterior domains. Theorem 3. Let 1 < p < p S Ω R N be a domain in R N. Then there exists C(N, p, m) > 0 independent of Ω u such that any nonnegative solution u in (1.5) satisfies (1.8) 2 D ν u(x) C(N, p, m)dist 2m (x, Ω), x Ω. In particular if Ω is an exterior domain, i.e. the set {x R N ; x > R} Ω for some R > 0, then 2 D ν u(x) C(N, p, m) x 2m, x 2R.

4 4 GUOZHEN LU, PEIYONG WANG AND JIUYI ZHU The Liouville-type theorems for semilinear elliptic equations in Euclidean space R N half space R N + are well-nown in literature. Among many results in [GS1], the Liouville-type theorem for the subcritical elliptic equation is shown. Namely, there is no nontrivial C 2 solution of the problem { u = u p in R (1.9) N, u 0 in R N, if N 3 1 < p < N+2. See also [CL] for a simpler proof with the Kelvin transform N 2 moving plane method of Gidas-Ni-Nirenberg [GNN]. In [DG], the authors consider the following mixed (Dirichlet-Neumann) boundary conditions in a half space, i.e. the problem (1.10) u = u p in R N +, u 0 in R N +, u = 0 on Γ 0 := {x R N + x N = 0, x 1 > 0}, u x N = 0 on Γ 1 := {x R N + x N = 0, x 1 < 0}, where N 3 1 < p < N+2. Using the idea of the Kelvin transform combined with the N 2 moving plane method, they prove the nonexistence of nontrivial solutions in (1.10) under the assumption that the solution is bounded. In next theorem, we are able to remove this boundedness assumption of the solution. In fact, our theorem is stated as: N+2 N 2 Theorem 4. If u C 2 (R N +) C(R N +) is a solution of (1.10) with N 3 1 < p <, then u 0. For the semilinear elliptic system, an important model is the Lane-Emden system: { u1 = u p 2 in R (1.11) N, u 2 = u q 1 in R N. It is conjectured that if 1 p q + 1 > n 2 n, then there are no nontrivial classical solutions of (1.11) in R N. The conjecture has been proved to be true for radial solutions in all dimensions in [M]. The cases of N = 3, 4 for the conjecture in general have been also solved recently in [PQS] [So] respectively. The interested reader can refer to the above papers references therein for detailed descriptions (see also the wors [BMa], [SZ], etc.). In [DS], the authors study the elliptic system (1.12) u 1 + u α u α α α = 0 in R N, α 1 1 α 2 1 u 2 + u α u α 2 1 = 0 in R N, prove that (1.12) does not have bounded positive classical solutions in R N if (1.13) 1 < α 1, α 2 < N + 2 N 2. We also show that the boundedness assumption of the solution for the elliptic system (1.12) is not necessary. Indeed, we can establish that

5 LIOUVILLE THEOREMS AND HIGHER ORDER ELLIPTIC EQUATIONS 5 Theorem 5. The classical positive solutions u 1 u 2 in (1.12) are trivial under the assumption of (1.13). We end our introduction by mentioning that nonexistence results of nonnegative solutions have been recently established by the first third authors [LZ] for certain classes of integral equations which are closely related to polyharmonic equations in half spaces by a completely different method from that used in the current paper, namely, the method of moving planes in integral form developed in [CLO]. The outline of the paper is as follows. Section 2 is devoted to obtaining the Liouvilletype theorem, then singularity decay estimates for higher order elliptic equations. In Section 3, the Liouville-type theorem with mixed boundary condition for semilinear elliptic equation is shown. The Liouville-type theorem for the elliptic systems is considered in Section 4. Throughout the paper, C c denote generic positive constants, which are independent of u may vary from line to line. 2. Higher order elliptic equations We state the following technical lemma used frequently in this paper. Lemma 1. (Doubling lemma) Let (X, d) be a complete metric space = D Σ X, with Σ closed. Define M : D (0, ) to be bounded on compact subsets of D fix a positive number. If y D is such that M(y)dist(y, Γ) > 2, where Γ = Σ \ D, then there exists x D such that M(x)dist(x, Γ) > 2, M(x) M(y) M(z) 2M(x), z D B(x, M 1 (x)). Remar 1. If Γ =, then dist(x, Γ) :=. The proof of the above lemma is given in [PQS]. Based on the doubling property, we can start the rescaling process to prove local estimates of solutions of superlinear problems. The idea is by a contradiction argument. One can argue that the local estimates of solutions are violated. By an appropriate rescaling, the blow up sequence of solutions converges to a bounded solution of a limiting equation in R N. Then from the non-existence of solutions in the limiting equation, we could infer that the local estimates exist. In this spirit, we conclude the proof of Theorem 1. We first state the lemma about the local a priori estimates for higher order elliptic equation (see [ADN] or [RW]). Let u satisfy the following equation (2.1) ( ) m u = g(x) in Ω. Then we have Lemma 2. Let Ω be a ball {x R N : x < R}. Suppose u W 2m,q satisfies (2.1). Then there exits a constant C > 0 depending only on Ω, N, m R such that for any σ (0, 1) C (2.2) u W 2m,q (Ω B σr ) (1 σ) ( g 2m L q (Ω) + u L q (Ω)).

6 6 GUOZHEN LU, PEIYONG WANG AND JIUYI ZHU The following lemma is the Liouville-type theorem for higher order elliptic equations with boundedness assumption in [S]. Lemma 3. The higher order elliptic equation (1.3) does not have positive solutions under assumption of (1.4) provided p < N 1+2m 2m + 1 < N (or N 2m + 1 N 1 2m p < ). proof of Theorem 1: Suppose that a solution u to the equation (1.3) is unbounded in the sense that (1.4) is violated. Namely, the following boundedness is not true sup u <, sup i u <, i = 1, 2,, m 1. Then, there exists a sequence of (y ) R N such that as. Set i u(y ) M(y) := ( i u(y) ) 2m : R N + R. Then M(y ) as. By taing D = Σ = X = R N + in the doubling lemma Remar 1 ( See also, e.g. [RW]), there exists another sequence of (x ) such that Define We introduce a new function M(x ) M(y ) M(z) 2M(x ), z B /M(x )(x ) R N +. d := x,n M(x ) H := {ξ R N ξ N > d }. v (ξ) := u(x + ξ ) M(x ) M 2m (x ) Then, v (ξ) is the nonnegative solution of { ( ) (2.3) m v = v p in H, v = v = = v = 0 on H satisfying (2.4) (2.5) i v (0) 2m+2()i = 1 i v (ξ) 2m+2()i 2, ξ H B (0). Two cases may occur as, either case (1) x,n M(x ).

7 LIOUVILLE THEOREMS AND HIGHER ORDER ELLIPTIC EQUATIONS 7 for a subsequence still denoted as before, or case (2) x,n M(x ) d for a subsequence still denoted as before, here d 0. If case (1) occurs, i.e. H B (0) R N as, then for any smooth compact D in R N, there exists 0 large enough such that D (H B (0)) as 0. By the classical W 2m,q estimates for higher order elliptic equation in Lemma 2 (2.5), we have v W 2m,q ( D) C for any 1 < q <. Therefore, we can extract a convergent subsequence v v in D, where v C 2,τ for some τ > 0. Furthermore, using a diagonal line argument, v v in C 2,τ loc (R N ) v solves ( ) m v = v p in R N. From Theorem A, u is trivial. However, (2.4) implies that i v(0) 2m+2()i = 1, which indicates that u is nontrivial. Obviously a contradiction is arrived. If case (2) occurs, we mae a further translation. Set Then ṽ satisfies (2.6) While (2.7) (2.8) ṽ (ξ) := v (ξ d e N ) for ξ R N +. { ( )mṽ = ṽ p in R N +, ṽ = ṽ = = ṽ = 0 on R N +. i ṽ (d e N ) 2m+2()i = 1 i ṽ (ξ) 2m+2()i 2, ξ R N + B (d e N ). Observe that (2.6) could also be reduced to a system of elliptic equations. Let Then, (w (0) (2.9) Furthermore,,, w() ) solves (ṽ,, ( ) ṽ ) := (w (0),, w() ). w () = (w (0) w (i) w (0) )p in R N +, = w (i+1) in R N +, i = 0, m 2, = = w () = 0 on R N +. w (i) w (i) (d e N ) 2m+2()i = 1 (ξ) 2m+2()i 2, ξ R N + B (d e N ).

8 8 GUOZHEN LU, PEIYONG WANG AND JIUYI ZHU For any smooth compact D in R N +, there exists 0 large enough such that D (R N + B (d e N ) for any > 0. By the classical elliptic estimates, w (i) (ξ) C 2,τ (D) C for some τ > 0. Thans to the Arzelá-Ascoli Theorem, there exists a function (v (0),, v () ) such that (w (0),, w() ) converges to (v (0),, v () ) in C 2 ( D). Through a diagonal line argument, (w (0),, w() ) converges in Cloc 2 (RN +) to (v (0),, v () ), which solves (2.10) satisfies (2.11) (2.12) v () = (v (0) ) p in R N +, v (i) = v (i+1) in R N +, i = 0, m 2, v (0) = = v () = 0 on R N +, v (i) (de N ) 2m+2()i = 1 v (i) (ξ) 2m+2()i 2, ξ R N +. From (2.10), v (0) actually solves { ( ) (2.13) m v (0) = (v (0) ) p in R N +, v (0) = v (0) = = v (0) = 0 on R N +. Hence v (0) 0 by lemma 3. Note that N 1+2m > N+2m. Then N 1 2m N 2m v(1),, v () 0, which contradicts (2.11). Therefore, the proof is completed. Next, we consider the singularity decay estimates of the higher order elliptic equation. The main idea is the same as the case of the semilinear elliptic equation in [PQS]. Proof of Theorem 2: Suppose (1.7) is not true. Then there exist sequences of Ω, u, y Ω such that u solves (1.5) on Ω the function satisfies M := 2 D ν u 2m+() ν M (y ) 2(1 + dist 1 (y, Ω)) 2dist 1 (y, Ω). Adapting from the doubling lemma, there exists x Ω such that M (x ) M (y ), M (x ) > 2dist 1 (x, Ω ) M (z) 2M (x ), if z x M 1 (x ).

9 LIOUVILLE THEOREMS AND HIGHER ORDER ELLIPTIC EQUATIONS 9 We introduce a new function Then, the function v solves v (ξ) := u (x + ξ ) M (x ) M 2m (x ), ξ. (2.14) ( ) m 2mp v (ξ) = f (v (ξ)) := M (x )f(m 2m (x )v ). Also, (2.15) (2.16) 2 2 D ν v 2m+() ν (0) = 1 By (1.6) the continuity of f, we have Furthermore, it follows that D ν v 2m+() ν (ξ) 2, ξ. c f(s) C(s p + 1), for s 0. 2mp cm (x ) f (v (ξ)) C, ξ. For any smooth compact D in R N, there exists 0 large enough such that D B (0) if 0. Using the classical estimate of the higher order elliptic equation in (2.2) (2.16), v (ξ) W 2m,q loc (R N ) for any 1 < q < v W 2m,q ( D) C. By the Sobolev imbedding a diagonal line argument, v (ξ) converges in C 2,τ loc (R N ) to a function v C 2,τ loc (R N ) for some τ > 0. Since v (ξ) is positive for ξ R N, then Consequently, v is a nonnegative solution of f (v ) ρv p as. ( ) m v(ξ) = ρv p (ξ), ξ R N. By an appropriate rescaling, v is trivial from Theorem A, which contradicts (2.15). Hence, the conclusion holds. Next, we turn to the special case of f(u) = u p. Proof of Theorem 3: Suppose (1.8) fails, Then, there exist sequences of Ω, u, y Ω such that u solves (1.5) in Ω the function satisfies M := 2 D ν u 2m+() ν M (y ) > 2dist 1 (y, Ω ).

10 10 GUOZHEN LU, PEIYONG WANG AND JIUYI ZHU By the doubling lemma, it follows that there exists x Ω such that M (x ) M (y ), M (x ) > 2dist 1 (x, Ω ) M (z) 2M (x ), if z x M 1 (x ). As before, we introduce a new function Then, the function v satisfies v (ξ) := u (x + ξ ) M (x ) M 2m (x ), ξ. (2.17) ( ) m v (ξ) = v p (ξ), ξ with (2.18) 2 2 D ν v 2m+() ν (0) = 1 D ν v 2m+() ν (ξ) 2, ξ. For any smooth compact D in R N, there exists 0 large enough such that D B (0) if 0. By the classical W 2m,q estimates for higher order elliptic equation in Lemma 2, we have v W 2m,q ( D) C for any 1 < q <. Furthermore, using the Sobolev imbedding Theorem a diagonal line argument, v v in C 2,τ loc (R N ) v solves ( ) m v = v p. With the aid of Theorem A, u is trivial. Nevertheless, from (2.18), it is impossible to be trivial. Hence, a contradiction is arrived. Then the theorem is verified. 3. Elliptic equation with mixed boundary condition In this section, we consider the Liouville-type theorem with mixed boundary condition. Lemma 4. Let u W 1,2 loc (RN +) C 0 (R N +) be a wea solution of (1.10). Then the only bounded solution is u 0. The lemma above is the Liouville-type theorem with boundedness assumptions of solutions in [DG]. Proof of Theorem 4: Suppose that there exists a unbounded solution u. Then, there exits a sequence (y ) R N + such that u(y ) as. Let M := u 2 : R N + R, then M(y ) as since p > 1. Following from the doubling lemma by taing D = Σ = X = R N + Remar 1, there exists another sequence of (x ) such that M(x ) M(y ) M(z) 2M(x ), z B /M(x )(x ) R N +.

11 Define LIOUVILLE THEOREMS AND HIGHER ORDER ELLIPTIC EQUATIONS 11 d := x,n M(x ) H := {ξ R N ξ N > d }. As in Theorem 1, we introduce a new function v (ξ) = u(x + ξ ) M(x ) M 2 (x ) Then, v (ξ) is the nonnegative solution of v = v p in H, v 0 in H, (3.1) v = 0 on Γ 0 := {ξ H ξ N = d, ξ 1 > 0}, v ξ N = 0 on Γ 1 := {ξ H ξ N = d, ξ 1 < 0} satisfying (3.2) v (0) 2 = 1 (3.3) v 2 (ξ) 2, ξ H B (0). As before, two cases may occur as, either case (1) x,n M(x ) for a subsequence still denoted as before, or case (2) x,n M(x ) d for a subsequence still denoted as before, here d 0. If case (1) occurs, i.e. H B (0) R N as, then for any smooth compact D in R N, there exists 0 large enough such that D (H B (0) as 0. By the classical elliptic equation estimates (3.3), we have v W 2,q ( D) C for any 1 < q <. Therefore, from the Sobolev embedding theorem, we can extract a convergent subsequence v that converges to v in D, where v C 1,τ ( D). Furthermore, using a diagonalization argument, v v in C 1,τ loc (RN ) v solves v = v p in R N. By the classical Liouville theorem of semilinear elliptic equation in Euclidean space, u is trivial, which contradicts (3.2). If case (2) occurs, we mae a further translation. Define Then ṽ satisfies (3.4) ṽ (ξ) := v (ξ d e N ) for ξ R N +. ṽ = ṽ p in R N +, ṽ 0 in R N +, ṽ = 0 on Γ 0 := {ξ R N + ξ N = 0, ξ 1 > 0}, ṽ ξ N = 0 on Γ 1 := {ξ R N + ξ N = 0, ξ 1 < 0}..

12 12 GUOZHEN LU, PEIYONG WANG AND JIUYI ZHU While, (3.5) ṽ 2 (d e N ) = 1 (3.6) ṽ 2 (ξ) 2, ξ R N + B (d e N ). For any smooth compact neighborhood D of the origin in R N +, we have {ṽ } is uniformly bounded in C 2,τ (K) C τ ( D) for some 0 < τ < 1 2, where K is a compact set of R N + with dist(k, Γ) > 0( The a priori estimates could be done in section 6 in [CP]) where Γ := {ξ R N + ξ N = 0, ξ 1 = 0}. By the Arzelá-Ascoli Theorem, ṽ converges to ṽ in C 2 (K) C( D). In addition, by a diagonalization argument, ṽ converges locally in R N + to ṽ C 2 (R N +) C(R N +) such that v satisfies ṽ = ṽ p in R N +, ṽ(ξ) 2 2 in R (3.7) N +, ṽ = 0 on Γ 0 = {ξ R N + ξ N = 0, ξ 1 > 0}, ṽ ξ N = 0 on Γ 1 = {ξ R N + ξ N = 0, ξ 1 < 0}. Deduced from Lemma 4, we have ṽ 0. However, from (3.5), ṽ(de N ) = 1. Clearly, a contradiction is achieved. Therefore, the proof Theorem 4 is complete. 4. Elliptic systems In the last section, we consider the elliptic systems. We could show that boundedness is removable in the Liouville theorem of (1.12). The following lemma is established in [DS]. Lemma 5. The system (1.12) does not have a bounded positive solution in R N, provided 1 < α 1, α 2 < N + 2 N 2. Proof of theorem 5. Suppose by contradiction that there exists a sequence of (y ) such that u 1 (y ) or u 2 (y ) as. Let M(y) := u α (y) + u α (y). Then, M(y ) as. From the doubling lemma by letting D = Σ = X = R N Remar 1, there exists a sequence of (x ) such that M(x ) M(y ) M(z) 2M(x ), if z x M 1 (x ). Let α = 2 α 1 1 β = 2 α 2 1. We rescale (u 1, u 2 ) by ξ M(x ) ) ṽ 1, (ξ) := u 1(x + M α (x )

13 Then, (ṽ 1,, ṽ 2, ) satisfies (4.1) LIOUVILLE THEOREMS AND HIGHER ORDER ELLIPTIC EQUATIONS 13 ṽ 2, (ξ) := u 2(x + M β (x ) ṽ 1, + ṽ α 1 ṽ 2, + ṽ α 2 ξ ) M(x ) α 1, ṽα 1 α 1 1 2, + ṽα 2. 2, = 0, α 1 1 α 2 1 1, = 0. Moreover ṽ 1 α 1, (0) + ṽ 1 β 2, (0) = 1 ṽ 1 α 1, (ξ) + ṽ 1 β 2, (ξ) 2, ξ. For any smooth compact D in R N, there exists 0 large enough such that D B (0) if 0. Using the classical W 2,q estimates for elliptic equation, we have 2 ṽ i, W 2,q (D) C i=1 for 1 < q <. By the stard Sobolev imbedding theorem, there exists a sequence of (ṽ 1,, ṽ 2, ) that converges to (ṽ 1, ṽ 2 ) in C 1,τ (D) for some τ > 0. Employing a diagonal line argument, we readily deduce that (ṽ 1,, ṽ 2, ) converges in C 1,τ loc (RN ) to a solution (ṽ 1, ṽ 2 ) in R N which satisfies (4.2) Furthermore, ṽ 1 + ṽ α ṽ α α α = 0, α 1 1 α 2 1 ṽ 2 + ṽ α ṽ α 2 1 = 0. ṽ 1 α 1 (0) + ṽ 1 β 2 (0) = 1. So (ṽ 1, ṽ 2 ) is nontrivial. This contradicts Lemma 5. Therefore, we then complete the proof of the theorem. Acnowledgement: Part of the wor was done when the first author was visiting Beijing Normal University in China. References [ADN] S. Agmon, A. Douglis L. Nirenberg, Esitmates near the boundary for solutions of elliptic partial differential equaition satisfying general boundary condtions, I. Comm. Pure Appl. Math. 12(1959), [BCN] H. Berestyci, L. Caffarelli L. Nirenberg, Further equalitative properties for elliptic equaiton in unbounded domains. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25(1997), [BM] I. Birindelli E. Mitidieri, Liouville theorems for ellptic inequalities applications, Proc. Roy. Soc. Edinburgh 128A(1998), [BMa] J. Busca R. Manasevich, A Liouville-type theorem for Lane-Emden systems, Indiana Univ. Math. J. 51 (2002), no. 1, [CDM] G. Caristi, L. D Ambrosio E. Mitidieri, Representation formulae for solutions to some classes of higher order systems related Liouville theorems. Milan J. Math. 76 (2008), [CL] W. Chen C. Li, Classification of solutions of some nonlinear nonlinear elliptic equations, Due Math. J. 63(1991),

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