A note on Liouville type theorem of elliptic inequality Δu+ u σ ≤0 on Riemannian manifolds
aa r X i v : . [ m a t h . DG ] O c t A NOTE ON LIOUVILLE TYPE THEOREM OF ELLIPTICINEQUALITY ∆ u + u σ ON RIEMANNIAN MANIFOLDS
HUI-CHUN ZHANG
Abstract.
Let σ > M be a complete Riemannian manifold.In a very recent work [10], Grigor ′ yan and Sun proved that a Liouvilletype theorem for nonnegative solutions of elliptic inequality( ∗ ) ∆ u ( x ) + u σ ( x ) , x ∈ M. via a pointwise condition of volume growth of geodesic balls. In thisnote, we improve their result to that an integral condition on volumegrowth implies the same uniqueness of ( ∗ ). It is inspired by the well-known Varopoulos-Grigor ′ yan’s criterion for parabolicity of M . Introduction
Let σ > M be a complete noncompact Riemannian manifoldwithout boundary. Consider the semilinear elliptic inequality(1.1) ∆ u ( x ) + u σ ( x ) , x ∈ M, where ∆ is the Laplace-Beltrami opertor on M . A function u ∈ W , ( M ) iscalled a weak solution of the inequality (1.1) if − Z M h∇ u, ∇ ψ i dµ + Z M u σ ψdµ ψ ∈ W , ( M ) with compact support.In Euclidean setting, i.e. M = R n , it has a long history to study theuniqueness of nonnegative solutions for (1.1) (or more general elliptic inequa-tions and equalities). There are many beautiful results have been obtainedin this subject. We refer the readers to, for instance, [1, 2, 3, 4, 11, 12, 13]and references therein for them. Many of these results are based on com-parison principle and careful choices of test functions for (1.1). To use thismethod on a manifold M , one have to estimate the second order derivativeof distance functions, which needs some assumptions on curvature of M .Surprisingly, in recent works Grigor ′ yan-Kondratiev [9] and Grigor ′ yan-Sun [10] proved a curvature-free Liouville type theorem for nonnegative weaksolution of (1.1) in terms of volume growth of geodesic balls in M as follows. Theorem 1.1 (Grigor ′ yan-Sun [10]) . Let M be a complete Riemannianmanifold without boundary. Fix a point x ∈ M and set V ( r ) := µ (cid:0) B ( x , r ) (cid:1) the volume of geodesic ball of radius r centered at x . Assume that, for some
C > , the inequality (1.2) V ( r ) Cr σσ − (ln r ) σ − holds for all large enough r . Then any nonnegative weak solution of (1.1) is identically equal to 0. They also showed that the exponents σσ − and σ − are sharp.On the other hand, let us recall that a manifold M is said to be parabolic if a Liouville type theorem holds for nonnegative solution of inequality∆ u ( x ) , x ∈ M, i.e., any nonnegative weak solution of ∆ u M must be constant.Cheng and Yau [5] proved that V ( r ) Cr , for some C >
0, is a sufficientcondition for parabolicity of M . Nowdays, a well-known sharp sufficient con-dition for parabolicity is the following integral condition, which was provedindependly by Varopoulos [14] and Grigor ′ yan [7, 8]: Z ∞ rV ( r ) dr = ∞ . Inspired by Varopoulos-Grigor ′ yan’s condition for the parabolicity of M ,we ask a natural question: what is a sufficient condition for Liouville typetheorem of inequlity (1.1) via an integral estimate of V ( r )? Of course, sucha condition should cover the above pointwise condition (1.2).In this remark, we solve this question. Our main result states as follows: Theorem 1.2.
Let M be a complete Riemannian manifold without bound-ary. Assume that (1.3) lim inf t → + t σσ − Z ∞ V ( r ) r σ − σ − + t dr < ∞ . Then any nonnegative weak solution of (1.1) is identically equal to 0.Remark . Condition (1.2) in Theorem 1.1 implies the condition (1.3). Infact, (1.2) = ⇒ t σσ − Z ∞ V ( r ) drr σ − σ − + t t σσ − Z ∞ (ln r ) σ − drr t = Γ( σσ − , where Γ( · ) is Gamma function.2. Proof of Theorem 1.2
Proof of Theorem 1.2.
Let u ∈ W , ( M ) be a nontrivial nonnegative solu-tion to the inequality (1.1).The proof of Theorem 1.1 in [10] contains two main parts. Firstly, theauthors derived a useful priori estimate in terms of a test function andpositive parameters (which will be recalled in Lemma 2.1 below). Secondly,they chose specific test functions to conclude R M u σ dµ = 0 . Our proof ofTheorem 1.2 is basically along the same line in [10]. The different from
IOUVILLE TYPE THEOREM FOR ELLIPTIC INEQUALITY 3
Grigor ′ yan-Sin’s proof will appear in the second part. We will choose avariation of their test functions to conclude R M u σ dµ = 0 . Firstly, let us recall the useful priori estimate given in [10]. We summarizeit as the following lemma:
Lemma 2.1 (Grigor ′ yan-Sun, [10]) . Set s = 8 σ/ ( σ − . Then there existsa constant C > such that the following property holds:For any t ∈ (cid:0) , min { , σ − } (cid:1) , any nonempty compact set K ⊂ M , and any Lipschitz function φ on M withconpact support such that φ on M and φ ≡ in a neighborhood of K , we have (2.1) Z M φ s u σ dµ C (cid:16) Z M \ K φ s u σ dµ (cid:17) t +12 σ · J ( t, φ ) and (2.2) (cid:16) Z M φ s u σ dµ (cid:17) − t +12 σ C · J ( t, φ ) , where J ( t, φ ) := t − − σ σ − (cid:16) Z M |∇ φ | σ − tσ − dµ (cid:17) · (cid:16) Z M |∇ φ | σσ − t − dµ (cid:17) σ − t − σ . Proof.
Inequality (2.1) is Eq.(2.10) in [10], and inequality (2.2) is Eq.(2.11)in [10]. (cid:3)
In the following, we will consider a family of specific test functions φ n ,which are modifications from original structures in [10].Fix any t ∈ (cid:0) , min { , σ − } (cid:1) . We set R = R ( t ) := exp(1 /t ). We considerthe function φ t ( x ) = , r ( x ) < R, (cid:16) r ( x ) R (cid:17) − t , r ( x ) > R, and a family of functions, for any n = 1 , , , · · · ,ξ t,n ( x ) = , r ( x ) n R, − r ( x )2 n R , n R r ( x ) n +1 R, , r ( x ) > n +1 R. Consider the functions(2.3) φ t,n ( x ) := φ t ( x ) · ξ t,n ( x ) . Then, for each n = 1 , , · · · , function φ t,n ( x ) is Lipschitz continuous on M and has compact support, and φ t,n ≡ B R ( t ) := B ( x , R ( t )). HUI-CHUN ZHANG
Claim:
There exists a constant C > such that, for any t ∈ (cid:0) , min { , σ − } (cid:1) with A ( t ) := Z ∞ V ( r ) r σ − σ − + t dr < ∞ , we have (2.4) lim sup n →∞ [ J ( t, φ t,n )] σ σ − t − C · t σσ − · A ( t ) . Proof of Claim:
In the proof, the parameter t is fixed. To simplify the no-tations, we denote by φ := φ t , ξ n := ξ t,n and φ n := φ t,n . Notice that ∇ φ n = ξ n · ∇ φ + φ · ∇ ξ n . We have |∇ φ n | ξ n · |∇ φ | + φ · |∇ ξ n | ;and, by the inequality ( A + B ) a a − ( A a + B a ) for all A, B > a > |∇ φ n | a σσ − − (cid:2) ξ an · |∇ φ | a + φ a · |∇ ξ n | a (cid:3) for any a ∈ [1 , σσ − ]. In the following, we denote by σ := 4 σσ − . Similar as in [10], we need to estimate the integral R M |∇ φ n | a dµ. For any a ∈ [1 , σ ], we have Z M |∇ φ n | a dµ σ − · (cid:16) Z M \ B R |∇ φ | a dµ + Z B n +1 R \ B nR φ a |∇ ξ n | a dµ (cid:17) := 2 σ − · (cid:0) I ( a ) + II ( a, n ) (cid:1) , (2.5)where B R := B ( x , R ), and we have used that ∇ φ = 0 in B R and that |∇ ξ n | supported in B n +1 R \ B n R .Before we estimate the above integrals I ( a ) and II ( a, n ), we need thefollowing simple (but important) observation: If the parameter a ∈ [1 , σ ] satisfies (2.6) a ( t + 1) > t + 2 σσ − . Then we have (2.7) ∞ X n =1 V (2 n R ) (cid:0) n − R (cid:1) a ( t +1) · σ · A ( t ) := C · A ( t ) . In particular, it implies that (2.8) lim n →∞ V (2 n R ) (cid:0) n − R (cid:1) a ( t +1) = 0 . IOUVILLE TYPE THEOREM FOR ELLIPTIC INEQUALITY 5
Indeed, we calculate directly to conclude ∞ X n =1 V (2 n R ) (cid:0) n − R (cid:1) a ( t +1) = 4 a ( t +1) · · ∞ X n =1 V (2 n R ) (cid:0) n +1 R (cid:1) a ( t +1) · n +1 R − n R n +1 R a ( t +1) · · ∞ X n =1 Z n +1 R n R V ( r ) drr a ( t +1)+1 · σ · Z ∞ V ( r ) drr a ( t +1)+1 , (2.9)we we have used that t < a σ and that R = exp(1 /t ) >
1. Combiningwith (2.6) and (2.9), we can obtain ∞ X n =1 V (2 n R ) (cid:0) n − R (cid:1) a ( t +1) · σ Z ∞ V ( r ) drr t + σσ − +1 = 2 · σ · A ( t ) . Ths is the desired estimate (2.7).Now let us estimate I ( a ). Assume that the parameter a satisfies (2.6), wehave I ( a ) = Z M \ B R |∇ φ | a dµ Z M \ B R h tR · (cid:16) rR (cid:17) − t − i a dµ = e a · t a Z M \ B R r a ( t +1) dµ (since R t = e )= e a · t a · ∞ X n =1 Z B nR \ B n − R r a ( t +1) dµ e a · t a · ∞ X n =1 V (2 n R ) (cid:0) n − R (cid:1) a ( t +1) e σ · C · t a A ( t ) (cid:0) by a σ and (2.7) (cid:1) . (2.10)Let us estimate II ( a, n ). Assume that the parameter a satisfies (2.6), wehave II ( a, n ) = Z B n +1 R \ B nR φ a |∇ ξ n | a dµ (cid:16) n RR ) − at (cid:16) n R (cid:17) a · V (2 n +1 R )= R at · V (2 n +1 R )(2 n R ) a ( t +1)) R t = e = e a · V (2 n +1 R )(2 n R ) a ( t +1)) . (2.11)Combining with (2.8), (2.11) and that a σ , we have(2.12) lim n →∞ II ( a, n ) = 0 . HUI-CHUN ZHANG
Therefore, according to (2.5),(2.10) and (2.12), we obtain, for any a ∈ [1 , σ ]satisfying (2.6),(2.13) lim sup n →∞ Z M |∇ φ n | a dµ σ − · e σ · C · t a A ( t ) := C · t a A ( t ) . We take a = 2 σ − tσ − a = 2 σσ − t − . Then it is easy to check that a , a satisfy (2.6). Indeed, a ( t + 1) = 2 σσ − t − t σ − > σσ − t (since t σ −
12 )and a ( t + 1) = 2 σσ − · σ − σ − t − · ( t + 1) > σσ − · ( t + 1) > σσ − t. Now, by using J ( t, φ n ) = t − − σ σ − (cid:16) Z M |∇ φ n | a dµ (cid:17) · (cid:16) Z M |∇ φ n | a dµ (cid:17) a and (2.13), we can conclude thatlim sup n →∞ J ( t, φ n ) t − − σ σ − · C + a · t a +1 · [ A ( t )] + a = C σ − t − σ · t + σ σ − − tσ − · [ A ( t )] σ − t − σ Then lim sup n →∞ [ J ( t, φ n )] σ σ − t − C · t ( + σ σ − − tσ − ) · σ σ − t − · A ( t )= C · t σσ − · (1 − t σ − t − ) · A ( t ) . (2.14)Noticing that lim t → + t − σσ − · t σ − t − = 1 , we have that the function t t − σσ − · t σ − t − is bounded on (0 ,
1) uniformly.Set the constant C := C · sup Claim is completed. (cid:3) Now let us continue the proof of Theorem 1.2.According to (1.3), there is a sequence of numbers { t α } ∞ α =1 , going to 0,such that(2.15) t σσ − α · A ( t α ) = t σσ − α Z ∞ V ( r ) r σ − σ − + t α dr C , ∀ α = 1 , , · · · for some constant C , independent of α. Without loss the generality, we canalso assume that t α ∈ (0 , min { , σ − } ) , for all α = 1 , , , · · · . IOUVILLE TYPE THEOREM FOR ELLIPTIC INEQUALITY 7 By using the above Claim , we have, for each α = 1 , , · · · ,(2.16) lim sup n →∞ J ( t α , φ t α ,n ) ( C · C ) σ − tα − σ max { ( C C ) σ − σ , } := C . In the following is similar as in [10]. We want to show u ∈ L σ ( M ), andmoreover R M u σ dµ = 0 . Fix arbitrary a nonempty compact set K ⊂ M .Notice that R ( t α ) = exp(1 /t α ) → ∞ as α → ∞ . So, we have K ⊂ B R ( t α ) for all large enough α . Hence, for any sufficient large α , φ t α ,n ≡ K holds for any n = 1 , , · · · . . For such α , we can apply Lemma 2.1 to t α , K and function φ t α ,n ; and we conclude that(2.17) Z K u σ dµ Z M φ st α ,n u σ dµ C (cid:16) Z M \ K φ st α ,n u σ dµ (cid:17) tα +12 σ · J ( t α , φ t α ,n )and(2.18) Z K u σ dµ Z M φ st α ,n u σ dµ (cid:16) C · J ( t α , φ t α ,n ) (cid:17) σ σ − tα − , for all n = 1 , , · · · , where we have used that φ t α ,n ≡ K .By combining (2.16) and (2.18), we obtain Z K u σ dµ (cid:16) C · C (cid:17) σ σ − tα − for all large enough α . Letting α → ∞ , we have(2.19) Z K u σ dµ (cid:16) C · C (cid:17) σ σ − := C . By combining with(2.17),(2.19),(2.16) and that φ t α ,n M , we have Z K u σ dµ C (cid:16) Z M \ K u σ dµ (cid:17) tα +12 σ · C , for all large enough α . Letting α → ∞ , we have(2.20) Z K u σ dµ C · C · (cid:16) Z M \ K u σ dµ (cid:17) σ . By using the arbitrariness of K , we can take K = B r for any r > 0. Com-bining with (2.19) and (2.20) and letting r → ∞ , we have Z M u σ dµ = 0 , which implies u ≡ M , and the proof of Theorem 1.2 is completed. (cid:3) Acknowledgements. We would like to thank Dr. Yuhua Sun for hisinteresting in the paper. The author is partially supported by GuangdongNatural Science Foundation S2012040007550 and by China Postdoctoral Sci-ence Foundation 2012T50736, 2012M521639. HUI-CHUN ZHANG References [1] S. Alarc´on; J. Garci´a-Meli´an & A. Quaas, Nonexistence of positive supersolutions tosome nonlinear elliptic porblems, J. Math. Pures Appl. 99(2013) 618–634.[2] Biduat-Veron, M.-F., & Pohozaev, S., Nonexistence results and estimates for somenonlinear elliptic problems , J. Anal. Math., 84 (2001), 1–49.[3] Caffarelli, L.; Garofalo, N. & Segala, F., A gradient bound for entire solutions of quasi-linear equatios and its consequences , Comm. Pure Appl. Math., 47 (1994), 1457–1473.[4] Caristi, G.; Mitidieri, E.& Pohozaev, S. I., Some Liouville theorems for quasilinearelliptic inequalities , Doklady Math. 79 (2009), no. 1, 118–124.[5] S. Y. Cheng & S. T. Yau, Differential equations on Riemannian manifolds and theirgeometric applications , Comm. Pure Appl. Math., 28(1975), no. 3, 333–354.[6] Gidas, B.& Spruck, J., Global and local behavior of positive solutions of nonlinearelliptic equations , Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.[7] A. Grigor ′ yan, On the existence of a Green function on a manifold (in Russian),Uspekhi Math. Nauk 38(1)(1983) 161–162, Engl. transl.: Russian Math. Surveys38(1)(1983) 190–191.[8] A. Grigor ′ yan, On the existence of positive fundamental solution of the Laplace equa-tion on Riemannian manifolds (in Russian), Maat. Sb. 128(3)(1985) 354–363, Engl.transl.: Math. USSR Sb. 56 (1987) 349–358.[9] A. Grigor ′ yan & V. A. Kondratiev, On the existence of positive solutions of semilinearelliptic inequalities on Riemannian manifolds . Around the research of Vladimir MazyaII, 203–218. International Mathematical Series (New York), 12. Springer, New York,2010.[10] A. Grigor ′ yan & Y. Sun, On Nonnegative Solutions of the Inequality ∆ u + u σ onRiemnannian Manifolds , to appear in Comm. Pure Appl. Math., (2013).[11] Pohozaev, S. I., Critical nonlinearities in partial differential equations , Milan J. Math.77 (2009), 127–150.[12] J. Serrin, Entire solutions of nonlinear Poisson equations , Proc. London Math. Soc.(3), 24 (1972), 348–366.[13] J. Serrin & H. Zou, Cauchy-Liouville and universal boundedness theorems for quasi-linear elliptic equations and inequalities , Acta Math., 189(2002), 79–142.[14] N. Varopoulos, The Poisson kernel on positively curved manifolds , J. Funct. Anal.44(1981), 359–380., J. Funct. Anal.44(1981), 359–380.