March  2019, 39(3): 1257-1268. doi: 10.3934/dcds.2019054

Fractional equations with indefinite nonlinearities

1. 

Department of Mathematical Sciences, Yeshiva University, New York NY 10033, USA

2. 

School of Mathematics, Shanghai Jiao Tong University, Shanghai, China

3. 

Department of Mathematics, Louisiana State University, Baton Rouge LA 70803, USA

* Corresponding author, partially supported by NSFC 11571233

Received  January 2018 Published  December 2018

Fund Project: The first author is partially supported by the Simons Foundation Collaboration Grant for Mathematicians 245486
The third author is partially supported by NSF DMS 1500468.

In this paper, we consider a fractional equation with indefinite nonlinearities
$(-\vartriangle )^{α/2} u = a(x_1) f(u) $
for
$0<α<2$
, where
$a$
and
$f$
are nondecreasing functions. We prove that there is no positive bounded solution. In particular, in the case
$a(x_1) = x_1$
and
$f(u) = u^p$
, this remarkably improves the result in [15] by extending the range of
$α$
from
$[1,2)$
to
$(0,2)$
, due to the introduction of new ideas, which may be applied to solve many other similar problems.
Citation: Wenxiong Chen, Congming Li, Jiuyi Zhu. Fractional equations with indefinite nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1257-1268. doi: 10.3934/dcds.2019054
References:
[1]

B. BarriosL. Del PezzzoJ. Garcá-Mellán and A. Quaas, A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions, Disc. Cont. Dyn. Sys., 37 (2017), 5731-5746.  doi: 10.3934/dcds.2017248.

[2]

H. BerestyckiI. Capuzzo-Dolcetta and L. Nirenberg, Supperlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonl. Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.

[3]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[4]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[5]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure. Appl. Math, 62 (2009), 597-638.  doi: 10.1002/cpa.20274.

[6]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 4603-4615.  doi: 10.3934/dcds.2018201.

[7]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[8]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4, 2010.

[9]

W. Chen and C. Li, Maximum principle for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

[10]

W. ChenC. Li and Y. Li, A direct method of moving planes for fractional Laplacian, Advances in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[11]

W. Chen, C. Li and Y. Li, A direct blow-up and rescaling argument on nonlocal elliptic equations, International J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.

[12]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Cal. Var. & PDEs, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.

[13]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[14]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[15]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Diff. Equa., 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[16]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Disc. Cont. Dyn. Sys. - Series A, 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.

[17]

J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 3939-3953.  doi: 10.3934/dcds.2018171.

[18]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, dvances in Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.

[19]

B. GidasW. Ni and L. Nirenberg, Symmetry and the related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[20]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[21]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.

[22]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.

[23]

S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Mat. Pura Appl., 195 (2016), 273-291.  doi: 10.1007/s10231-014-0462-y.

[24]

Y. Lei, Asymptotic properties of positive solutions of the Hardy Sobolev type equations, J. Diff. Equa., 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.

[25]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy Littlewood Sobolev system of integral equations, Cal. Var. & PDEs, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.

[26]

C. Li, Z. Wu and H. Xu, Maximum Principles and Bocher Type Theorems, Proceedings of the National Academy of Sciences, June 20, 2018.

[27]

C. S. Lin, On Liouville theorem and apriori estimates for the scalar curvature equations, Ann. Scula Norm. Sup. Pisa CI.Sci., 27 (1998), 107-130. 

[28]

B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 5339-5349.  doi: 10.3934/dcds.2018235.

[29]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.

[30]

G. Lu and J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.

[31]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var. & PDEs, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.

[32]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.

[33]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[34]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. de Math. Pures et Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[35]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[36]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

show all references

References:
[1]

B. BarriosL. Del PezzzoJ. Garcá-Mellán and A. Quaas, A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions, Disc. Cont. Dyn. Sys., 37 (2017), 5731-5746.  doi: 10.3934/dcds.2017248.

[2]

H. BerestyckiI. Capuzzo-Dolcetta and L. Nirenberg, Supperlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonl. Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.

[3]

C. BrandleE. ColoradoA. de Pablo and U. Sanchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proc Royal Soc. of Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[4]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[5]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure. Appl. Math, 62 (2009), 597-638.  doi: 10.1002/cpa.20274.

[6]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 4603-4615.  doi: 10.3934/dcds.2018201.

[7]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[8]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series, vol. 4, 2010.

[9]

W. Chen and C. Li, Maximum principle for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

[10]

W. ChenC. Li and Y. Li, A direct method of moving planes for fractional Laplacian, Advances in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[11]

W. Chen, C. Li and Y. Li, A direct blow-up and rescaling argument on nonlocal elliptic equations, International J. Math., 27 (2016), 1650064, 20 pp. doi: 10.1142/S0129167X16500646.

[12]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Cal. Var. & PDEs, 56 (2017), Art. 29, 18 pp. doi: 10.1007/s00526-017-1110-3.

[13]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[14]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[15]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Diff. Equa., 260 (2016), 4758-4785.  doi: 10.1016/j.jde.2015.11.029.

[16]

T. Cheng, Monotonicity and symmetry of solutions to frac- tional Laplacian equation, Disc. Cont. Dyn. Sys. - Series A, 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.

[17]

J. Dou and Y. Li, Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 3939-3953.  doi: 10.3934/dcds.2018171.

[18]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, dvances in Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.

[19]

B. GidasW. Ni and L. Nirenberg, Symmetry and the related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[20]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Diff. Equa., 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.

[21]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.

[22]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. H. Poincare Nonl. Anal., 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.

[23]

S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Mat. Pura Appl., 195 (2016), 273-291.  doi: 10.1007/s10231-014-0462-y.

[24]

Y. Lei, Asymptotic properties of positive solutions of the Hardy Sobolev type equations, J. Diff. Equa., 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.

[25]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy Littlewood Sobolev system of integral equations, Cal. Var. & PDEs, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.

[26]

C. Li, Z. Wu and H. Xu, Maximum Principles and Bocher Type Theorems, Proceedings of the National Academy of Sciences, June 20, 2018.

[27]

C. S. Lin, On Liouville theorem and apriori estimates for the scalar curvature equations, Ann. Scula Norm. Sup. Pisa CI.Sci., 27 (1998), 107-130. 

[28]

B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys. - Series A, 38 (2018), 5339-5349.  doi: 10.3934/dcds.2018235.

[29]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.

[30]

G. Lu and J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.

[31]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Cal. Var. & PDEs, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.

[32]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.

[33]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[34]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. de Math. Pures et Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[35]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[36]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

[1]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[2]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059

[3]

Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015

[4]

Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051

[5]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201

[6]

Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235

[7]

Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166

[8]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154

[9]

Yuxia Guo, Shaolong Peng. Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1637-1648. doi: 10.3934/cpaa.2022037

[10]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure and Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048

[11]

Jérôme Coville, Nicolas Dirr, Stephan Luckhaus. Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks and Heterogeneous Media, 2010, 5 (4) : 745-763. doi: 10.3934/nhm.2010.5.745

[12]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure and Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125

[13]

Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065

[14]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

[15]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209

[16]

Pablo Amster, Mariel Paula Kuna, Dionicio Santos. Stability, existence and non-existence of $ T $-periodic solutions of nonlinear delayed differential equations with $ \varphi $-Laplacian. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022070

[17]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769

[18]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949

[19]

Henrik Garde, Stratos Staboulis. The regularized monotonicity method: Detecting irregular indefinite inclusions. Inverse Problems and Imaging, 2019, 13 (1) : 93-116. doi: 10.3934/ipi.2019006

[20]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1031-1051. doi: 10.3934/cpaa.2009.8.1031

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (524)
  • HTML views (152)
  • Cited by (7)

Other articles
by authors

[Back to Top]