LIOUVILLE THEOREM AND GRADIENT ESTIMATES FOR NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS

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1 Electronic Journal of Differential Equations, Vol. 017 (017), No. 58, pp ISSN: URL: or LIOUVILLE THEOREM AND GRADIENT ESTIMATES FOR NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS WEN WANG, HUI ZHOU, XINQUAN ZHANG Counicated by Gioanni Molica Bisci Abstract. In this article we study a nonlinear elliptic equation by using the axiu principle and cutoff functions, We establish related gradient estiates, the Liouille theore, and the Harnack inequality. 1. Introduction and stateent of ain results In 1981, Gidas-Spruck 3 deried the following result. Theore 1.1. Let M n be a coplete anifold with nonnegatie Ricci curature. Assue that h(x) C (M n ) and α > 0 satisfy the following conditions: (1) h(x) 0 on M n ; () h(x) 0 on M n ; (3) for r(x) large, log h(x) C/r(x) and if n 4, h(x) C(r(x)) σ with σ n3, where r(x) is the geodesic distance between x and soe fixed point p; (4) 1 α n n. If u(x) is a nonnegatie solution of then u(x) 0. u hu α = 0, For α = 1, Li-Yau 9 deonstrated the sae result under the condition that h(x) = o(r(x)) as r(x). Later, Li 6 proed that as 1 α n n (n 4), the condition (3) of Theore 1.1 is unnecessary. On these conditions were further weakened, see 1, 5, 7. In 010, Yang 11 studied the equation u cu α = 0 on a noncopact coplete Rieannian anifold, where α > 0 and c are two real constants. The corresponding gradient estiates and Liouille type theore are also deried. 010 Matheatics Subject Classification. 58J35, 35K05, 53C1. Key words and phrases. Gradient estiate; nonlinear elliptic equation; Liouille theore; Harnack inequality. c 017 Texas State Uniersity. Subitted October 8, 016. Published October 16,

2 W. WANG, H. ZHOU, X. ZHANG EJDE-017/58 Recently, Wang 10 deduced gradient estiates and Liouille type theore for positie solutions to the equation f u cu = 0 on sooth easure space with -Bakry-Éery curature bounded by Ric f, ( 1)K, where K 0. Inspired by the works 3, 8, 10, 11, we inestigate the nonlinear elliptic equation u λ(x)u l = 0, > 1 (1.1) on a coplete Rieannian anifold with Ricci curature bounded below, where > 1 and l are real nubers, and λ(x) C (M n ). If M n = Ω is a bounded sooth doain in R n and λ(x) 0 is a constant, the equation (1.1) is regarded as the thin fil equation, which depict a steady state of the thin fil. Concrete content can be seen 4. Our ain results reads as follows. Theore 1.. Let (M n, g) be a coplete Rieannian anifold without boundary. Suppose that B R is a geodesic ball of radius R around p M and Ric(B R ) K with K 0. Also pose that there exist two positie nubers δ and τ such that λ(x, t) δ and λ τ λ. Let u(x) is a positie solution to the equation (1.1) 1 u1. and = (a) Assue that l 1, then B p(r) (b) Assue that l < 1, then B p(r) C 4( 1) 1 1 R (1 KR) K τ H 1. (1.) C 4( 1) 1 1 R (1 KR) K τ H. (1.3) Where C 4 is a constant depending only on n, and H 1 = H = ( 1)(n 1) ( 1) ( l 1) δ ( 1) n 1 ( 1)(n 1) ( 1) ( l 1) δ ( 1) n 1 ( 1 ( 1 1, inf 1. Moreoer, if (M n, g) has nonnegatie Ricci curature, letting R, we hae following estiate for l 1, and for l < 1, Where C(, n, l, K, δ, τ, ), (1.4) C(, n, l, K, δ, τ, ). (1.5) C(, n, l, K, δ, τ, ) = C 4( 1) (K τ) H 1, 1 C (, n, l, K, δ, τ,, inf ) = C 4( 1) 1 (K τ) H. By using (1.4) and (1.5), we derie the related Harnack inequalities.

3 EJDE-017/58 LIOUVILLE THEOREM AND GRADIENT ESTIMATES 3 Corollary 1.3. Let (M n, g) be a noncopact coplete Rieannian anifold without boundary. Suppose that Ric(M n ) 0. Let u(x) is a positie solution of the equation (1.1), and = 1 u1. If l 1, then C(, n, l, K, δ, τ, (x) (y) exp r(x, y) ). (1.6) inf If l < 1, then (x) (y) exp r(x, y) C (, n, l, K, δ, τ,, inf ). (1.7) inf Theore 1.4. Let (M n, g) be a coplete Rieannian anifold without boundary. Suppose that B R is a geodesic ball of radius R around p M and Ric(B R ) K with K 0. Let u(x) is a positie solution of the equation (1.1). Let = 1 u1 and λ τ λ for soe positie constant τ. If λ 0 and l (n1)(1) λ 0 and l (n1)(1), then we hae 1 C 4 B p(r) R (1 KR) K τ, (1.8) where C 4 is a constant depending only on n. Letting R, then we infer on a coplete noncopact Rieannian anifold, C 4 (K τ). (1.9) Applying (1.9), we can derie the following Liouille type theore as λ(x) is a constant. Corollary 1.5. Let (M n, g) be a noncopact coplete Rieannian anifold without boundary. Suppose that Ric(M n ) 0 and u(x) is a positie solution of the equation (1.1), where λ(x) is a constant. If λ 0 and l (n1)(1) or λ 0 and l (n1)(1), then u is a constant. Theore 1.6. Let (M n, g) be a coplete noncopact Rieannian anifold with Ric(M n ) K with K 0. Let u(x) is a positie solution to the equation u λu l = 0, (1.10) where λ > 0 is a constant. Let = 1 u1 and 1 l (n1)(1). If λ then for any x M n, ( 1)(n 1)K ( 1 ( 1 ) l 1, (1.11) (1) n1 ( l 1) ( 1) (n 1) K ( 1) ( 1)(n 1) ( 1) ( 1) n 1 ( l 1) λ ( 1 1. or

4 4 W. WANG, H. ZHOU, X. ZHANG EJDE-017/58 If ( 1)(n 1)K ( 1 λ ( 1 ) l 1, (1) n1 ( l 1) then ust be a constant. Note that by taking l = 1 in Theore 1.6, our result partially generalize Wang s result in 10. By (1.11), we can find the lower bound estiate as l, and the upper bound estiate as l for positie solutions of (1.10).. Preliinaries To proe our ain results, we need the lea below. Let = 1 u1, then ( 1) = λ( 1) ( 1 l (.1) Lea.1. Let (M n, g) be a coplete Rieannian anifold without boundary. Suppose that B R is a geodesic ball of radius R around p M and Ricci(B R ) K with K 0. Let u(x) is a positie solution to the equation (1.1) and = 1 u1. Let w = and G = ϕw, where ϕ(x) is a sooth cutoff function (see the proof of Theore 1.). Suppose that G(x) reaches the axiu alue at x 0 and ϕ(x 0 ) > 0. Then at x 0, ϕ w n (n 1) (n 1)( 1) 1 G 1 ϕ 1 n n 1 ϕ (n 1)( 1) ϕ G n (n 1) ( 1) (n 1)( 1) l 1 ( 1 λ 1 G 1 ϕ λ ( 1 1 n 1 λ( 1 ϕ 1. n 1 λ( 1 ϕ ϕ G (n 1)KG ) l1 1 ϕ Proof. After calculations we obtain that ( ) w = = 4 3 = Hess Ric((, )) 4 3. Since G reaches at the axiu at x 0, so we hae G = 0. Then at x 0, (.) (.3) w = G ϕ ϕ, (.4) = G ϕ ϕ G. (.5) ϕ

5 EJDE-017/58 LIOUVILLE THEOREM AND GRADIENT ESTIMATES 5 Choose an orthonoral frae {e 1, e,, e n } around x 0, such that e 1 =. Then 4 = 1j, (.6) On the other hand, we hae Hess 11 1α 11 = 11 = n n 1 11 = n n n 1 j=1 = 11. (.7) αα 1α 1 ( n n 1 αα ) 1α 1 n 1 ( 11) 1α n n 1 ( ) 1α 11 1 n 1 1 w λ( w λ( 1 1. Putting (.6) and (.7) into (.8), and using (.5), we hae Hess Ric(, ) n n α 11 1 n n 1 1 w λ( 1 n 1j 11 1 n 1 n 1 j=1 1 w λ( 1 ) l1 1 (n 1)K 1 w λ( n 1 1 w λ( 1 = n n n n 1 1 w λ( 1 n G G = 4(n 1) ϕ ϕ ϕ 1 n 1 1 G (n 1) ϕ G ϕ ϕ 1 (n 1)K 1 ) l1 1 (n 1)K w λ( 1 ) l1 1 (n 1)K 1 1 w λ( 1 1 w λ( 1 ) l1 1 ) l1 ) l1 1 ) l1 1 1 (.8)

6 6 W. WANG, H. ZHOU, X. ZHANG EJDE-017/58 = n 4(n 1) (G) n G ϕ 4(n 1) ϕ 4 ϕ w (n 1)( 1) w (n 1)( 1) λ( n 1 λ( 1 ) 1 G (n 1) ϕ ϕ (n 1)K l1 1 n (n 1) G ϕ ϕ 3 1 G (n 1)( 1) ϕ w 1 G n 1 ϕ λ( 1 w 1 1 G (n 1) ϕ ϕλ( 1 1 ) l1 1 n (G) n ϕ G = 4(n 1) ϕ 4(n 1) ϕ ϕ n G ϕ (n 1) ϕ ϕ 1 (G) λ ( 1 (n 1)( 1) ϕ (n 1)( 1) G 1 ϕ λ ( 1 n 1 1 (G) 1 (n 1)( 1) ϕ 1 n 1 λ( 1 G 1 ϕ 1 G ϕ (n 1)( 1) ϕ ϕ 1 n 1 λ( 1 ϕ 1 (n 1)K ϕ = n 4(n 1) G (n 1)( 1) ϕ n (n 1) 1 G (n 1)( 1) ϕ ϕ n ϕ 1 4(n 1) ϕ 3 G (n 1)( 1) λ( 1 G 1 ϕ 1 n 1 λ( 1 ϕ 1 λ ( 1 ϕ n 1 1 = (n 1)K. Noting that > 0 when > 1. Using the aboe inequality in (.3) and applying (.1) and (.5), then (.) can be inferred. 3. The proof of ain results Proof of Theore 1.. Construct a sooth function θ(t) : 0, ) 0, 1 { 1, 0 t 1 θ(t) = 0, t > such that C 1 θ θ 0, θ C θ. (3.1) Define the sooth cutoff function ϕ : M R by ϕ(x) = θ( r(x) R ). We pose that G = ϕw = ϕ attains its axial alue at x 0 B R. We can pose that G(x 0 ) > 0, because otherwise the proof is triial. Then at x 0, we hae G = ϕ w ϕ w ϕ w = ϕ w G ϕ ϕ ϕ w

7 EJDE-017/58 LIOUVILLE THEOREM AND GRADIENT ESTIMATES 7 Note that ϕ = θ R θ r R Since G 0 is alid at x 0, we hae = ϕ ϕ G G ϕ ϕ ϕ w. ϕ = θ r R, θ R (n 1)(1 KR)θ R. 0 θ (n 1)(1 KR)θ 3n 4 θr G θr (n 1) (θ ) R θ G ( 1)(n n ) 4 G ( 1)(n ) G G θ ( 1) (n 1) θ (n 1)( 1) Rθ θ ( 1) ( 1)(n 1) l 1 ( λ 1 1 G 1 (n 1)KG θ λ ( 1 1 λ ( 1 n 1 1 G θ R θ λ ( 1 n 1 θ 1. (3.) Applying the inequality ax bx b 4a with a > 0, we hae λ ( 1 n 1 θ 1 λ ( 1 n 1 1 G θ R θ G(θ ) (n 1)R θ, λ ( 1 n 1 ) (n 1) λ λ G. By the Cauchy inequality, it follows that l1 θ 1 ϕ λ ( 1 1 (3.3) G G θ Rθ θ G θ G(θ ) R θ. (3.4) Substituting (3.3) and (3.4) into (3.), we obtain θ (n 1)(1 KR)θ 0 θr θr G 3n (θ ) (n 1) R θ G ( 1) G ( 1)(n ) G(θ ) ( 1) (n 1) θ ( 1)(n 1) R θ (n 1)KG (n 1) λ λ G ( 1) ( 1)(n 1) l 1 1 ( 1 λ G 1. (3.5)

8 8 W. WANG, H. ZHOU, X. ZHANG EJDE-017/58 Fro (3.1) and λ τλ, we hae C (n 1)(1 KR)C1 ( 1) G 0 G R θr ( 1) (n 1) θ 1 n C 1 ( 1)(n 1) R G (n 1)KG (n 1)τG θ ( 1) ( 1)(n 1) l 1 λ ( 1 1 G 1 C (n 1)(1 KR)C1 ( 1) G G R θr ( 1) (n 1) θ C1 1 R G (n 1)KG (n 1)τG θ ( 1) ( 1)(n 1) l 1 λ ( 1 1 G 1. Multiply by θ to both side of (3.6), and using 0 θ 1 we obtain for l 1 ( 1) 0 ( 1) (n 1) G C1 1 R C (n 1)(1 KR)C1 R R G (n 1)K (n 1)τ G ( 1) ( 1)(n 1) l 1 1 Meanwhile, for l < 1 we obtain G λ ( 1 ( 1) 0 ( 1) (n 1) G C1 1 R G C (n 1)(1 KR)C1 R R G (n 1)K (n 1)τ G x M ( 1) ( 1)(n 1) l 1 1 ( 1 λ 1 G. inf 1 G. (3.6) (3.7) (3.8) We obsere that C (n 1)(1 KR)C1 R R C 3 R (1 KR), (3.9) for soe constant C 3 depending only on n. On the other hand, for the equation Ax Bx 0 with A > 0, B > 0, we hae x B A. By utilize the equation to (3.7) and (3.8), and noting (3.9) we obtain at the the axiu point x 0 for l 1 w(x) ϕw(x 0 ) = G(x 0 ) B p(r)

9 EJDE-017/58 LIOUVILLE THEOREM AND GRADIENT ESTIMATES 9 and for l < 1, ( 1)(n 1) C1 1 R ( 1) (n 1) ( 1) ( 1) (n 1) (K τ) ( 1) w(x) ϕw(x 0 ) = G(x 0 ) B p(r) ( 1)(n 1) ( 1) ( l 1) λ ( 1) n 1 ( 1)(n 1) C1 1 R ( 1) (n 1) ( 1) ( 1) (n 1) (K τ) ( 1) ( 1)(n 1) ( 1) ( l 1) λ ( 1) n 1 C 3 R (1 KR) ( 1 C 3 1, R (1 KR) ( 1 inf 1. The proof is coplete. (1) (1)(n1) l1 The proof of Theore 1.4. Siple calculations show that 1 0 as λ 0 and l (n1)(1) or λ 0 and l (n1)(1). Hence, dropping the last ter in (3.6) which is nonnegatie, we hae C (n 1)(1 0 R θr KR)C1 C1 1 R G (n 1)(K τ)g. θ ( 1) G G ( 1) (n 1) θ Multiplying by θ on both sides, and using 0 θ 1, we obtain ( 1) 0 ( 1) (n 1) G C1 1 R G C (n 1)(1 KR)C1 R R G (n 1)(K τ)g. x M Therefore, at the the axiu point x 0 we obtain w(x) ϕw(x 0 ) = G(x 0 ) B p(r) ( 1)(n 1) C1 1) R ( 1) (n 1) ( 1) ( 1) (n 1) (K τ), ( 1) C 3 R (1 KR) where we used (3.9). The proof is coplete.

10 10 W. WANG, H. ZHOU, X. ZHANG EJDE-017/58 The proof of Theore 1.6. It is not difficult to find that for 1 l (n1)(1). Then we hae for (3.6), ( 1) 0 ( 1) (n 1) G C1 1 R G C (n 1)(1 KR)C1 R R G (n 1)K (n 1)τ G ( 1) ( 1)(n 1) l 1 1 ( 1 λ By (3.10), and (3.9) we obtain at the the axiu point x 0, w(x) ϕw(x 0 ) = G(x 0 ) B p(r) ( 1)(n 1) C1 1 R ( 1) (n 1) K ( 1) ( 1) (n 1) ( 1) ( 1)(n 1) ( 1) ( l 1) λ ( 1) n 1 (1) (1)(n1) l G. C 3 (3.10) R (1 KR) ( 1 Letting R, we infer as ( 1)(n 1)K ( 1 1 λ ( ) l 1 (1), n1 ( l 1) and ( 1) (n 1) K ( 1) ( 1)(n 1) ( 1) ( l 1) λ ( 1) n 1 ( 1 On the other hand, as ( 1)(n 1)K ( 1 1 λ ( ) l 1 (1), n1 ( l 1) we derie that ust be constant Proof of Corollary 1.3. Let inial geodesic γ(s) : 0, 1 M n, so that γ(0) = y, γ(1) = x, then ln (x) (y) = r(x, y) d ln((γ(s))) ds = 1 γ ds = r(x, y) (γ(s)) γ (γ(s)) ds 1 0 (γ(s)) ds C(, n, l, K, δ, τ, ) ds inf

11 EJDE-017/58 LIOUVILLE THEOREM AND GRADIENT ESTIMATES 11 = r(x, y) C(, n, l, K, δ, τ, x M ). inf Acknowledgents. This work was ported by the Higher School Natural Science Foundation of Anhui Proince (KJ016A310, KJ017A937), by the Higher School outstanding young talent port project of Anhui proince in 017 (gxyq ), by the Acadeic Research Project of Hefei Noral Uniersity (017QN41,017QN44), and by the Natural Science Foundation of Anhui Proince ( MA16). We are grateful to Professor Jiayu Li for his encourageent. We also thank Professor Qi S Zhang for introduction of this subject in the suer course. We thank the anonyous referee for suggestions and references. References 1 S. Asserda; A Liouille theore forthe Schrodinger operator with drift, C. R. Acad. Sci. Paris, Ser. I, 34 (006), G. Bonanno, G. Molica Bisci V. Radulescu; Nonlinear elliptic probles on Rieannian anifolds and applications to Eden-Fowler type equations, Manuscripta Math. 14 (013), B. Gidas, J. Spruck; Global andlocal behaior of positie solutions of nonlinear elliptic equations, Co. Pure Appl. Math., 34 (1981), Z. Guo, J. Wei; Hausdoff diension of ruptures for solutions of a seilinear equation with singular nonlinearity, Manuscript Math., 10, (006), A. Melas; A Liouille type theore for the Schrödinger operator, Proc. Aer. Math. Soc., 17 (1999), G. Molica Bisci; Variational probles on the sphere, Recent Trends in Nonlinear Partial Differential Equations. Stationary probles, Contep. Math. 595 (013), E. Negrin; Gradient estiates and a Liouille type theore for the Schrodinger operator, J. Funct. Anal., 17 (1995), J. Y. Li; Gradient estiates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equationson Rieannian anifolds, J. Funct. Anal., 100 (1991), P. Li, S. T. Yau; On the parabolic kernel of the Schrodinger operator, Acta Math., 156 (1986), L. F. Wang; Liouille theores and gradient estiates for a nonlinear elliptic equation, J. Differential Equations, 60 (1) (016), Y. Y. Yang; Gradient Estiates for the Equation u cu α = 0 on Rieannian Manifolds, Acta Matheatica Sinica, English Series, 6 (6) (010), Wen Wang (corresponding author) School of Matheatics and Statistics, Hefei Noral Uniersity, Hefei 30601, China. School of atheatical Science, Uniersity of Science and Technology of China, Hefei 3006, China E-ail address: wwen014@ail.ustc.edu.cn Hui Zhou (corresponding author) School of Matheatics and Statistics, Hefei Noral Uniersity, Hefei 30601, China. School of atheatical Science, Uniersity of Science and Technology of China, Hefei 3006, China E-ail address: zhouhui0309@16.co Xinquan Zhang School of Matheatics and Statistics, Hefei Noral Uniersity, Hefei 30601, China E-ail address: @qq.co