# American Institute of Mathematical Sciences

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. Barrios, L. Del Pezzzo, J. 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. Berestycki, I. 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. Brandle, E. Colorado, A. 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. Chen, Y. 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. Chen, C. 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. Chen, C. 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. Chen, C. 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. Gidas, W. 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. Han, G. 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. Hang, X. 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. Lei, C. 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. Zhuo, W. Chen, X. 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. Barrios, L. Del Pezzzo, J. 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. Berestycki, I. 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. Brandle, E. Colorado, A. 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. Chen, Y. 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. Chen, C. 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. Chen, C. 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. Chen, C. 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. Gidas, W. 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. Han, G. 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. Hang, X. 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. Lei, C. 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. Zhuo, W. Chen, X. 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.
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