A priori estimates of solutions to nonlinear fractional Laplacian equation

Tao Zhang; Tingzhi Cheng; Tao Zhang; Tingzhi Cheng

doi:10.3934/era.2023056

Electronic Research Archive

2023, Volume 31, Issue 2: 1119-1133. doi: 10.3934/era.2023056

Previous Article Next Article

Research article

A priori estimates of solutions to nonlinear fractional Laplacian equation

Tao Zhang ¹,
Tingzhi Cheng ^{2
,
,}

1.
School of Mathematics and Information Science, Yantai University, Yantai 264005, China
2.
School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

Received: 08 September 2022 Revised: 03 December 2022 Accepted: 07 December 2022 Published: 20 December 2022

In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems.
- a priori estimates,
- fractional Laplacian,
- nonlocal problems
Citation: Tao Zhang, Tingzhi Cheng. A priori estimates of solutions to nonlinear fractional Laplacian equation[J]. Electronic Research Archive, 2023, 31(2): 1119-1133. doi: 10.3934/era.2023056

Related Papers:

Abstract

In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems.

References

[1]	A. Chang, M. J. Gursky, P. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differ. Equations, 1 (1993), 205–229. https://doi.org/10.1007/BF01191617 doi: 10.1007/BF01191617
[2]	Z. C. Han, Prescribing Gaussian curvature on $\mathbb{S}^{2}$, Duke Math. J., 61 (1990), 679–703. https://doi.org/10.1215/S0012-7094-90-06125-3 doi: 10.1215/S0012-7094-90-06125-3
[3]	R. Schoen, D. Zhang, Prescribed scalar curvature on then-sphere, Calc. Var. Partial Differ. Equations, 4 (1996), 1–25. https://doi.org/10.1007/bf01322307 doi: 10.1007/bf01322307
[4]	Y. Y. Li, Prescribing scalar curvature on $\mathbb{S}^{n}$ and related problems, Part I, J. Differ. Equations, 120 (1995), 319–410. https://doi.org/10.1006/jdeq.1995.1115 doi: 10.1006/jdeq.1995.1115
[5]	Y. Y. Li, Prescribing scalar curvature on $\mathbb{S}^{n}$ and related problems, Part II: existence and compactness, Commun. Pure Appl. Math., 49 (1996), 541–597. https://doi.org/10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A
[6]	B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equations, 6 (1981), 883–901. https://doi.org/10.1080/03605308108820196 doi: 10.1080/03605308108820196
[7]	C. C. Chen, C. S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes, Commun. Pure Appl. Math., 50 (1997), 971–1017. https://doi.org/10.1002/(SICI)1097-0312(199710)50:10<971::AID-CPA2>3.0.CO;2-D doi: 10.1002/(SICI)1097-0312(199710)50:10<971::AID-CPA2>3.0.CO;2-D
[8]	C. C. Chen, C. S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes, II, J. Differ. Geom., 49 (1998), 115–178. https://doi.org/10.4310/jdg/1214460938 doi: 10.4310/jdg/1214460938
[9]	W. Chen, C. Li, A priori estimates for prescribing scalar curvature equations, Ann. Math., 145 (1997), 547–564. https://doi.org/10.2307/2951844 doi: 10.2307/2951844
[10]	Y. Y. Li, L. Zhang, Compactness of solutions to the Yamabe problem, C.R. Math., 338 (2004), 693–695. https://doi.org/10.1016/j.crma.2004.02.018 doi: 10.1016/j.crma.2004.02.018
[11]	Y. Y. Li, L. Zhang, Compactness of solutions to the Yamabe problem, II, Calc. Var. Partial Differ. Equations, 24 (2005), 185–237. https://doi.org/10.1007/s00526-004-0320-7 doi: 10.1007/s00526-004-0320-7
[12]	Y. Y. Li, L. Zhang, Compactness of solutions to the Yamabe problem, III, J. Funct. Anal., 245 (2007), 438–474. https://doi.org/10.1016/j.jfa.2006.11.010 doi: 10.1016/j.jfa.2006.11.010
[13]	A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, in CISM International Centre for Mechanical Sciences, Courses and Lectures, Springer-Verlag, 378 (1997). https://doi.org/10.1007/978-3-7091-2664-6
[14]	A. A. Lokshin, Y. V. Suvorova, Mathematical theory of wave propagation in media with memory, Moskov. Gos. Univ., Moscow, 1982.
[15]	T. L. Jin, Y. Y. Li, J. G. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111–1171. https://doi.org/10.4171/JEMS/456 doi: 10.4171/JEMS/456
[16]	A. Chang, M. Gonz$\acute{a}$lez, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410–1432. https://doi.org/10.1016/j.aim.2010.07.016 doi: 10.1016/j.aim.2010.07.016
[17]	C. R. Graham, M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89–118. https://doi.org/10.1007/s00222-002-0268-1 doi: 10.1007/s00222-002-0268-1
[18]	X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275–302. https://doi.org/10.1016/j.matpur.2013.06.003 doi: 10.1016/j.matpur.2013.06.003
[19]	W. Chen, C. Li, Y. Li, A direct blow up and rescaling agrument on nonlocal elliptic equation, Int. J. Math., 27 (2016), 1650064. https://doi.org/10.1142/S0129167X16500646 doi: 10.1142/S0129167X16500646
[20]	L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equations, 32 (2007), 1245–1260. https://doi.org/10.1080/03605300600987306 doi: 10.1080/03605300600987306
[21]	X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Henri Poincare C, 31 (2014), 23–53. https://doi.org/10.1016/J.ANIHPC.2013.02.001 doi: 10.1016/J.ANIHPC.2013.02.001
[22]	X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052–2093. https://doi.org/10.1016/j.aim.2010.01.025 doi: 10.1016/j.aim.2010.01.025
[23]	A. Capella, J. Dávila, L. Dupaigne, Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Commun. Partial Differ. Equations, 36 (2011), 1353–1384. https://doi.org/10.1080/03605302.2011.562954 doi: 10.1080/03605302.2011.562954
[24]	S. Dipierro, G. Palatucci, E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, preprint, arXiv: 1202.0576.
[25]	R. Servadei, E. Valdinoci, Mountain Pass solutions for nonlocal elliptic operators, J. Math. Anal. Appl., 389 (2012), 887–898. https://doi.org/10.1016/j.jmaa.2011.12.032 doi: 10.1016/j.jmaa.2011.12.032
[26]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67–112. https://doi.org/10.1002/cpa.20153 doi: 10.1002/cpa.20153
[27]	L. Zhang, C. Li, W. Chen, T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbb {R}^n_+$, Discrete Contin. Dyn. Syst., 36 (2016), 1721–1736. https://doi.org/10.3934/dcds.2016.36.1721 doi: 10.3934/dcds.2016.36.1721
[28]	R. Zhuo, W. Chen, X. Cui, Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125–1141. https://doi.org/10.3934/dcds.2016.36.1125 doi: 10.3934/dcds.2016.36.1125
[29]	A. D. Aleksandrov, A charcteristic property of spheres, Ann. Mat. Pura Appl., 58 (1962), 303–315. https://doi.org/10.1007/BF02413056 doi: 10.1007/BF02413056
[30]	J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304–318. https://doi.org/10.1007/BF00250468 doi: 10.1007/BF00250468
[31]	B. Gidas, W. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209–243. https://doi.org/10.1007/BF01221125 doi: 10.1007/BF01221125
[32]	B. Gidas, W. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb {R}^n$, Adv. Math. Suppl. Stud. A, 7 (1981), 369–402.
[33]	H. Berestycki, L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237–275. https://doi.org/10.1016/0393-0440(88)90006-X doi: 10.1016/0393-0440(88)90006-X
[34]	H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat.-Bull. Braz. Math. Soc., 22 (1991), 1–37. https://doi.org/10.1007/BF01244896 doi: 10.1007/BF01244896
[35]	W. Chen, C. Li, Classifications of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615–622. https://doi.org/10.1215/S0012-7094-91-06325-8 doi: 10.1215/S0012-7094-91-06325-8
[36]	C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Commun. Partial Differ. Equations, 16 (1991), 491–526. https://doi.org/10.1080/03605309108820766 doi: 10.1080/03605309108820766
[37]	C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. Partial Differ. Equations, 16 (1991), 585–615. https://doi.org/10.1080/03605309108820770 doi: 10.1080/03605309108820770
[38]	W. Chen, C. Li, Y. Li, A direct method of moving planes for the fractional laplacian, Adv. Math., 308 (2017), 404–437. https://doi.org/10.1016/j.aim.2016.11.038 doi: 10.1016/j.aim.2016.11.038
[39]	D. G. de Figueiredo, P. L. Lions, R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41–63. https://doi.org/10.1007/978-3-319-02856-9_11 doi: 10.1007/978-3-319-02856-9_11
[40]	N. S. Landkoff, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)