Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems

Jacques Giacomoni; Deepak Kumar; Konijeti Sreenadh

doi:10.1515/acv-2021-0040

Artikel

Interior and boundary regularity results for strongly nonhomogeneous p,q-fractional problems

Jacques Giacomoni , Deepak Kumar und Konijeti Sreenadh

Veröffentlicht/Copyright: 25. September 2021

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Advances in Calculus of Variations Band 16 Heft 2

Abstract

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional ( p , q ) -Laplacian, denoted by ( - Δ ) p s 1 + ( - Δ ) q s 2 for s 2 , s 1 ∈ ( 0 , 1 ) and 1 < p , q < ∞ . We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional ( p , q ) -problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish a new Hopf-type maximum principle and a strong comparison principle in both situations.

Keywords: nonhomogeneous nonlocal operator; singular nonlinearity; local and boundary Hölder continuity; maximum principle; strong comparison principle

MSC 2010: 35J60; 35R11; 35B45; 35D30

Communicated by Juha Kinnunen

Acknowledgements

The authors thank both anonymous referees for the careful reading of this paper and for their remarks and comments, which have improved the initial version of our work.

References

[1] Adimurthi, J. Giacomoni and S. Santra, Positive solutions to a fractional equation with singular nonlinearity, J. Differential Equations 265 (2018), no. 4, 1191–1226. 10.1016/j.jde.2018.03.023Suche in Google Scholar

[2] V. Ambrosio and T. Isernia, On a fractional p & q Laplacian problem with critical Sobolev–Hardy exponents, Mediterr. J. Math. 15 (2018), no. 6, Paper No. 219. 10.1007/s00009-018-1259-9Suche in Google Scholar

[3] V. Ambrosio and V. D. Rădulescu, Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl. (9) 142 (2020), 101–145. 10.1016/j.matpur.2020.08.011Suche in Google Scholar

[4] R. Arora, J. Giacomoni and G. Warnault, Regularity results for a class of nonlinear fractional Laplacian and singular problems, NoDEA Nonlinear Differential Equations Appl. 28 (2021), no. 3, Paper No. 30. 10.1007/s00030-021-00693-9Suche in Google Scholar

[5] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 62. 10.1007/s00526-018-1332-zSuche in Google Scholar

[6] M. Bhakta and D. Mukherjee, Multiplicity results for ( p , q ) fractional elliptic equations involving critical nonlinearities, Adv. Differential Equations 24 (2019), no. 3–4, 185–228. 10.57262/ade/1548212469Suche in Google Scholar

[7] S. Biagi, S. Dipierro, E. Valdinoci and E. Vecchi, Semilinear elliptic equations involving mixed local and nonlocal operators, Proc. Roy. Soc. Edinburgh Sect. A (2021), 10.1017/prm.2020.75. 10.1017/prm.2020.75Suche in Google Scholar

[8] L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math. 304 (2017), 300–354. 10.1016/j.aim.2016.03.039Suche in Google Scholar

[9] L. Brasco, E. Lindgren and A. Schikorra, Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case, Adv. Math. 338 (2018), 782–846. 10.1016/j.aim.2018.09.009Suche in Google Scholar

[10] L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2016), no. 4, 323–355. 10.1515/acv-2015-0007Suche in Google Scholar

[11] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lect. Notes Unione Mat. Ital. 20, Springer, Cham, 2016. 10.1007/978-3-319-28739-3Suche in Google Scholar

[12] X. Cabré, S. Dipierro and E. Valdinoci, The Bernstein technique for integrodifferential equations, preprint (2020), https://arxiv.org/abs/2010.00376. Suche in Google Scholar

[13] L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp. 7, Springer, Heidelberg (2012), 37–52. 10.1007/978-3-642-25361-4_3Suche in Google Scholar

[14] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 59–88. 10.1007/s00205-010-0336-4Suche in Google Scholar

[15] A. Canino, L. Montoro, B. Sciunzi and M. Squassina, Nonlocal problems with singular nonlinearity, Bull. Sci. Math. 141 (2017), no. 3, 223–250. 10.1016/j.bulsci.2017.01.002Suche in Google Scholar

[16] W. Chen, S. Mosconi and M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018), no. 11, 3065–3114. 10.1016/j.jfa.2018.02.020Suche in Google Scholar

[17] Z.-Q. Chen, P. Kim, R. Song and Z. Vondraček, Sharp Green function estimates for Δ + Δ α / 2 in C 1 , 1 open sets and their applications, Illinois J. Math. 54 (2010), no. 3, 981–1024. 10.1215/ijm/1336049983Suche in Google Scholar

[18] C. De Filippis and G. Mingione, On the regularity of minima of non-autonomous functionals, J. Geom. Anal. 30 (2020), no. 2, 1584–1626. 10.1007/s12220-019-00225-zSuche in Google Scholar

[19] L. M. Del Pezzo and A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations 263 (2017), no. 1, 765–778. 10.1016/j.jde.2017.02.051Suche in Google Scholar

[20] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1279–1299. 10.1016/j.anihpc.2015.04.003Suche in Google Scholar

[21] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Suche in Google Scholar

[22] J. Fernández Bonder, A. Salort and H. Vivas, Interior and up to the boundary regularity for the fractional g-Laplacian: The convex case, preprint (2020), https://arxiv.org/abs/2008.05543. Suche in Google Scholar

[23] M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl. 37, Oxford University, Oxford, 2008. 10.1093/oso/9780195334722.003.0002Suche in Google Scholar

[24] J. Giacomoni, D. Goel and K. Sreenadh, Regularity results on a class of doubly nonlocal problems, J. Differential Equations 268 (2020), no. 9, 5301–5328. 10.1016/j.jde.2019.11.009Suche in Google Scholar

[25] J. Giacomoni, D. Kumar and K. Sreenadh, Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems, Calc. Var. Partial Differential Equations 60 (2021), no. 3, Paper No. 121. 10.1007/s00526-021-01994-8Suche in Google Scholar

[26] J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 1, 117–158. 10.2422/2036-2145.2007.1.07Suche in Google Scholar

[27] J. Giacomoni, I. Schindler and P. Takáč, Singular quasilinear elliptic systems and Hölder regularity, Adv. Differential Equations 20 (2015), no. 3–4, 259–298. 10.57262/ade/1423055202Suche in Google Scholar

[28] D. Goel, D. Kumar and K. Sreenadh, Regularity and multiplicity results for fractional ( p , q ) -Laplacian equations, Commun. Contemp. Math. 22 (2020), no. 8, Article ID 1950065. 10.1142/S0219199719500652Suche in Google Scholar

[29] J. Hernández, F. J. Mancebo and J. M. Vega, Nonlinear singular elliptic problems: Recent results and open problems, Nonlinear Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl. 64, Birkhäuser, Basel (2005), 227–242. 10.1007/3-7643-7385-7_12Suche in Google Scholar

[30] A. Iannizzotto, S. J. N. Mosconi and N. Papageorgiou, On the logistic equation for the fractional p-Laplacian, preprint (2021), https://arxiv.org/abs/2101.05535. Suche in Google Scholar

[31] A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32 (2016), no. 4, 1353–1392. 10.4171/rmi/921Suche in Google Scholar

[32] A. Iannizzotto, S. J. N. Mosconi and M. Squassina, Fine boundary regularity for the degenerate fractional p-Laplacian, J. Funct. Anal. 279 (2020), no. 8, Article ID 108659. 10.1016/j.jfa.2020.108659Suche in Google Scholar

[33] E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of integro-PDEs, J. Differential Equations 212 (2005), no. 2, 278–318. 10.1016/j.jde.2004.06.021Suche in Google Scholar

[34] S. Jarohs, Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv. Nonlinear Stud. 18 (2018), no. 4, 691–704. 10.1515/ans-2017-6039Suche in Google Scholar

[35] D. Kumar, V. D. Rădulescu and K. Sreenadh, Singular elliptic problems with unbalanced growth and critical exponent, Nonlinearity 33 (2020), no. 7, 3336–3369. 10.1088/1361-6544/ab81edSuche in Google Scholar

[36] T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), no. 3, 1317–1368. 10.1007/s00220-015-2356-2Suche in Google Scholar

[37] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. 10.1016/0362-546X(88)90053-3Suche in Google Scholar

[38] S. A. Marano and S. J. N. Mosconi, Some recent results on the Dirichlet problem for ( p , q ) -Laplace equations, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 2, 279–291. 10.3934/dcdss.2018015Suche in Google Scholar

[39] P. Mironescu and W. Sickel, A Sobolev non embedding, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 3, 291–298. 10.4171/rlm/707Suche in Google Scholar

[40] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), no. 3, 275–302. 10.1016/j.matpur.2013.06.003Suche in Google Scholar

[41] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J. 55 (2006), no. 3, 1155–1174. 10.1512/iumj.2006.55.2706Suche in Google Scholar

Received: 2021-04-22

Accepted: 2021-08-09

Published Online: 2021-09-25

Published in Print: 2023-04-01

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/acv-2021-0040

Schlagwörter für diesen Artikel

nonhomogeneous nonlocal operator; singular nonlinearity; local and boundary Hölder continuity; maximum principle; strong comparison principle