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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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