Home On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems
Article
Licensed
Unlicensed Requires Authentication

On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems

  • Prashanta Garain , Wontae Kim and Juha Kinnunen EMAIL logo
Published/Copyright: October 27, 2023

Abstract

We establish existence results for a class of mixed anisotropic and nonlocal p-Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, the Harnack inequality for solutions and the weak Harnack inequality for supersolutions.


Communicated by Matthias Hieber


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.2018.03.023Search in Google Scholar

[2] C. O. Alves and A. Moussaoui, Existence and regularity of solutions for a class of singular ( p ( x ), q ( x ) ) -Laplacian systems, Complex Var. Elliptic Equ. 63 (2018), no. 2, 188–210. 10.1080/17476933.2017.1298589Search in Google Scholar

[3] C. O. Alves, C. A. Santos and T. W. Siqueira, Uniqueness in W l o c 1 , p ( x ) ( Ω ) and continuity up to portions of the boundary of positive solutions for a strongly-singular elliptic problem, J. Differential Equations 269 (2020), no. 12, 11279–11327. 10.1016/j.jde.2020.08.038Search in Google Scholar

[4] R. Arora and V. D. Radulescu, Combined effects in mixed local-nonlocal stationary problems, preprint (2021), https://arxiv.org/abs/2111.06701. Search in Google Scholar

[5] K. Bal, P. Garain and T. Mukherjee, On an anisotropic p-Laplace equation with variable singular exponent, Adv. Differential Equations 26 (2021), no. 11–12, 535–562. 10.57262/ade026-1112-535Search in Google Scholar

[6] B. Barrios, I. De Bonis, M. Medina and I. Peral, Semilinear problems for the fractional Laplacian with a singular nonlinearity, Open Math. 13 (2015), no. 1, 390–407. 10.1515/math-2015-0038Search in Google Scholar

[7] M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys. 54 (2003), 771–783. 10.1007/s00033-003-3209-ySearch in Google Scholar

[8] M. Belloni and B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p , ESAIM Control Optim. Calc. Var. 10 (2004), no. 1, 28–52. 10.1051/cocv:2003035Search in Google Scholar

[9] S. Biagi, S. Dipierro, E. Valdinoci and E. Vecchi, Mixed local and nonlocal elliptic operators: regularity and maximum principles, Comm. Partial Differential Equations 47 (2022), no. 3, 585–629. 10.1080/03605302.2021.1998908Search in Google Scholar

[10] S. Biagi, S. Dipierro, E. Valdinoci and E. Vecchi, A Faber–Krahn inequality for mixed local and nonlocal operators, preprint (2021), https://arxiv.org/abs/2104.00830. Search in Google Scholar

[11] S. Biagi, S. Dipierro, E. Valdinoci and E. Vecchi, A Hong–Krahn–Szegö inequality for mixed local and nonlocal operators, Math. Eng. 5 (2023), no. 1, Paper No. 014. 10.3934/mine.2023014Search in Google Scholar

[12] S. Biagi, D. Mugnai and E. Vecchi, A Brezis–Oswald approach for mixed local and nonlocal operators, preprint (2021), https://arxiv.org/abs/2103.11382. 10.1142/S0219199722500572Search in Google Scholar

[13] S. Biagi, E. Vecchi, S. Dipierro and E. Valdinoci, Semilinear elliptic equations involving mixed local and nonlocal operators, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 5, 1611–1641. 10.1017/prm.2020.75Search in Google Scholar

[14] K. Biroud, Mixed local and nonlocal equation with singular nonlinearity having variable exponent, J. Pseudo-Differ. Oper. Appl. 14 (2023), no. 1, Paper No. 13. 10.1007/s11868-023-00509-7Search in Google Scholar

[15] T. Biset, B. Mebrate and A. Mohammed, A boundary-value problem for normalized Finsler infinity-Laplacian equations with singular nonhomogeneous terms, Nonlinear Anal. 190 (2020), Article ID 111588. 10.1016/j.na.2019.111588Search in Google Scholar

[16] L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), no. 6, 581–597. 10.1016/0362-546X(92)90023-8Search in Google Scholar

[17] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations 37 (2010), no. 3–4, 363–380. 10.1007/s00526-009-0266-xSearch in Google Scholar

[18] V. Bögelein, F. Duzaar and N. Liao, On the Hölder regularity of signed solutions to a doubly nonlinear equation, J. Funct. Anal. 281 (2021), no. 9, Paper No. 109173. 10.1016/j.jfa.2021.109173Search in Google Scholar

[19] 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.2018.09.009Search in Google Scholar

[20] 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-2015-0007Search in Google Scholar

[21] S. Buccheri, J. V. da Silva and L. H. de Miranda, A system of local/nonlocal p-Laplacians: The eigenvalue problem and its asymptotic limit as p , Asymptot. Anal. 128 (2022), no. 2, 149–181. 10.3233/ASY-211702Search in Google Scholar

[22] S.-S. Byun and E. Ko, Global C 1 , α regularity and existence of multiple solutions for singular p ( x ) -Laplacian equations, Calc. Var. Partial Differential Equations 56 (2017), no. 3, Paper No. 76. 10.1007/s00526-017-1152-6Search in Google Scholar

[23] 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.2017.01.002Search in Google Scholar

[24] A. Canino, B. Sciunzi and A. Trombetta, Existence and uniqueness for p-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 2, Article ID 8. 10.1007/s00030-016-0361-6Search in Google Scholar

[25] J. Carmona and P. J. Martínez-Aparicio, A singular semilinear elliptic equation with a variable exponent, Adv. Nonlinear Stud. 16 (2016), no. 3, 491–498. 10.1515/ans-2015-5039Search in Google Scholar

[26] Z.-Q. Chen, P. Kim, R. Song and Z. Vondraček, Boundary Harnack principle for Δ + Δ α / 2 , Trans. Amer. Math. Soc. 364 (2012), no. 8, 4169–4205. 10.1090/S0002-9947-2012-05542-5Search in Google Scholar

[27] Y. Chu, Y. Gao and W. Gao, Existence of solutions to a class of semilinear elliptic problem with nonlinear singular terms and variable exponent, J. Funct. Spaces 2016 (2016), Article ID 9794739. 10.1155/2016/9794739Search in Google Scholar

[28] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2013. 10.1137/1.9781611972597Search in Google Scholar

[29] M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), no. 2, 193–222. 10.1080/03605307708820029Search in Google Scholar

[30] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 4, 493–516. 10.1016/s0294-1449(98)80032-2Search in Google Scholar

[31] C. De Filippis and G. Mingione, Gradient regularity in mixed local and nonlocal problems, Math. Ann. (2022), 10.1007/s00208-022-02512-7. 10.1007/s00208-022-02512-7Search in Google Scholar

[32] 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.2011.12.004Search in Google Scholar

[33] E. DiBenedetto, C 1 + α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. 10.1016/0362-546X(83)90061-5Search in Google Scholar

[34] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1998. Search in Google Scholar

[35] Y. Fang, Existence, uniqueness of positive solution to a fractional Laplacians with singular nonlinearity, preprint (2014), https://arxiv.org/abs/1403.3149. Search in Google Scholar

[36] Y. Fang, B. Shang and C. Zhang, Regularity theory for mixed local and nonlocal parabolic p-Laplace equations, J. Geom. Anal. 32 (2022), no. 1, Paper No. 22. 10.1007/s12220-021-00768-0Search in Google Scholar

[37] C. Farkas, A. Fiscella and P. Winkert, Singular Finsler double phase problems with nonlinear boundary condition, Adv. Nonlinear Stud. 21 (2021), no. 4, 809–825. 10.1515/ans-2021-2143Search in Google Scholar

[38] C. Farkas and P. Winkert, An existence result for singular Finsler double phase problems, J. Differential Equations 286 (2021), 455–473. 10.1016/j.jde.2021.03.036Search in Google Scholar

[39] M. Foondun, Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part, Electron. J. Probab. 14 (2009), no. 11, 314–340. 10.1214/EJP.v14-604Search in Google Scholar

[40] P. Garain, On a class of weighted anisotropic p-Laplace equation with singular nonlinearity, Nonlinear Anal. 61 (2023), no. 2, 775–799. 10.12775/TMNA.2022.037Search in Google Scholar

[41] P. Garain, On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem, Manuscripta Mathematica (2023), 10.1007/s00229-023-01504-4. 10.1007/s00229-023-01504-4Search in Google Scholar

[42] P. Garain, On a degenerate singular elliptic problem, Math. Nachr. 295 (2022), no. 7, 1354–1377. 10.1002/mana.201900431Search in Google Scholar

[43] P. Garain, On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity, J. Geom. Anal. 33 (2023), no. 7, Paper No. 212. 10.1007/s12220-023-01262-5Search in Google Scholar

[44] P. Garain and J. Kinnunen, On the regularity theory for mixed local and nonlocal quasilinear parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), https://doi.org/10.2422/2036-2145.202110_006, to appear. 10.2422/2036-2145.202110_006Search in Google Scholar

[45] P. Garain and J. Kinnunen, On the regularity theory for mixed local and nonlocal quasilinear elliptic equations, Trans. Amer. Math. Soc. 375 (2022), no. 8, 5393–5423. 10.1090/tran/8621Search in Google Scholar

[46] P. Garain and J. Kinnunen, Weak Harnack inequality for a mixed local and nonlocal parabolic equation, J. Differential Equations 360 (2023), 373–406. 10.1016/j.jde.2023.02.049Search in Google Scholar

[47] P. Garain and E. Lindgren, Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations, Calc. Var. Partial Differential Equations 62 (2023), no. 2, Paper No. 67. 10.1007/s00526-022-02401-6Search in Google Scholar

[48] P. Garain and T. Mukherjee, Quasilinear nonlocal elliptic problems with variable singular exponent, Commun. Pure Appl. Anal. 19 (2020), no. 11, 5059–5075. Search in Google Scholar

[49] P. Garain and A. Ukhlov, Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems, Nonlinear Anal. 223 (2022), Paper No. 113022. 10.1016/j.na.2022.113022Search in Google Scholar

[50] 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/9780195334722.003.0002Search in Google Scholar

[51] A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), no. 3, 721–730. 10.1090/S0002-9939-1991-1037213-9Search in Google Scholar

[52] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), no. 1–2, 795–826. 10.1007/s00526-013-0600-1Search in Google Scholar

[53] S. E.-H. Miri, On an anisotropic problem with singular nonlinearity having variable exponent, Ric. Mat. 66 (2017), no. 2, 415–424. 10.1007/s11587-016-0309-5Search in Google Scholar

[54] T. Mukherjee and K. Sreenadh, On Dirichlet problem for fractional p-Laplacian with singular non-linearity, Adv. Nonlinear Anal. 8 (2019), no. 1, 52–72. 10.1515/anona-2016-0100Search in Google Scholar

[55] N. S. Papageorgiou and A. Scapellato, Positive solutions for anisotropic singular ( p , q ) -equations, Z. Angew. Math. Phys. 71 (2020), no. 5, Paper No. 155. 10.1007/s00033-020-01385-7Search in Google Scholar

[56] A. M. Salort and E. Vecchi, On the mixed local-nonlocal Hénon equation, Differential Integral Equations 35 (2022), no. 11–12, 795–818. 10.57262/die035-1112-795Search in Google Scholar

[57] B. Shang and C. Zhang, Hölder regularity for mixed local and nonlocal p-Laplace parabolic equations, Discrete Contin. Dyn. Syst. 42 (2022), no. 12, 5817–5837. 10.3934/dcds.2022126Search in Google Scholar

[58] C. Xia, On a class of anisotropic problems, Dissertation, Albert-Ludwigs-Universität Freiburg, 2012. Search in Google Scholar

[59] Q. Zhang, Existence and asymptotic behavior of positive solutions to p ( x ) -Laplacian equations with singular nonlinearities, J. Inequal. Appl. 2007 (2007), Article ID 19349. 10.1155/2007/19349Search in Google Scholar

Received: 2023-04-25
Published Online: 2023-10-27
Published in Print: 2024-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2023-0151/html
Scroll to top button