Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
This article is divided into two parts. In the first part, we examine the Brezis–Oswald problem involving a mixed anisotropic and nonlocal p-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and...
Saved in:
| Published in | Nonlinear analysis Vol. 256; p. 113796 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.07.2025
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0362-546X |
| DOI | 10.1016/j.na.2025.113796 |
Cover
| Abstract | This article is divided into two parts. In the first part, we examine the Brezis–Oswald problem involving a mixed anisotropic and nonlocal p-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and nonlocal p-Laplace equations, we prove a Sturmian comparison theorem, establish comparison and nonexistence results, derive a weighted Hardy-type inequality, and analyze a system of singular mixed anisotropic and nonlocal p-Laplace equations. A key component of our approach is the use of the Picone identity, which we adapt from the local and nonlocal cases. In the second part of the article, we focus on regularity estimates. In the elliptic setting, we establish a weak Harnack inequality and semicontinuity results. We also consider a class of doubly nonlinear mixed anisotropic and nonlocal parabolic equations, proving semicontinuity results and analyzing the pointwise behavior of solutions. These results rely on appropriate energy estimates, De Giorgi-type lemmas, and positivity expansions. Finally, we derive various energy estimates, which may be of independent interest. |
|---|---|
| AbstractList | This article is divided into two parts. In the first part, we examine the Brezis–Oswald problem involving a mixed anisotropic and nonlocal p-Laplace operator. We establish results on existence, uniqueness, boundedness, and the strong maximum principle. Additionally, for certain mixed anisotropic and nonlocal p-Laplace equations, we prove a Sturmian comparison theorem, establish comparison and nonexistence results, derive a weighted Hardy-type inequality, and analyze a system of singular mixed anisotropic and nonlocal p-Laplace equations. A key component of our approach is the use of the Picone identity, which we adapt from the local and nonlocal cases. In the second part of the article, we focus on regularity estimates. In the elliptic setting, we establish a weak Harnack inequality and semicontinuity results. We also consider a class of doubly nonlinear mixed anisotropic and nonlocal parabolic equations, proving semicontinuity results and analyzing the pointwise behavior of solutions. These results rely on appropriate energy estimates, De Giorgi-type lemmas, and positivity expansions. Finally, we derive various energy estimates, which may be of independent interest. |
| ArticleNumber | 113796 |
| Author | Garain, Prashanta |
| Author_xml | – sequence: 1 givenname: Prashanta surname: Garain fullname: Garain, Prashanta email: pgarain92@gmail.com organization: Department of Mathematical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, Odisha 760010, India |
| BookMark | eNp1kMtOwzAQRb0oEm1hzzI_kOBHYrfsUMVLqsQCkNhZjjORXFw72G6hP8O3kjRlyWo0M_dczdwZmjjvAKErgguCCb_eFE4VFNOqIISJJZ-gKWac5lXJ38_RLMYNxpgIxqfo58VvIfvcKWuSSmYPmXLN0Lv0N-iC7yAkAzHzbfYF6iOL3u6S8S5myWdb8w1Nj5noUy81-mjRn2S9VnbwisYaBypkYK3p0knR-F1tD0fhuO1UULW3_Rp66Oh_gc5aZSNcnuocvd3fva4e8_Xzw9Pqdp1rIkTKgQvQdFHWgrZ02T9d1SWjApZcVLyqGy5wKcSCNVSotq1K1oDGCjO6wJViFWFzhEdfHXyMAVrZBbNV4SAJlkOmciOdkkOmcsy0R25GBPq79gaCjNqA09CYADrJxpv_4V_iv4bH |
| Cites_doi | 10.1515/acv-2015-0007 10.1090/S0002-9939-2014-12294-5 10.1007/s00033-003-3209-y 10.1016/j.jfa.2021.109173 10.1007/s00028-022-00832-4 10.1016/j.jfa.2014.05.023 10.1515/forum-2023-0151 10.1007/s00526-022-02378-2 10.1090/S0002-9947-08-04544-3 10.1007/s10958-024-07141-w 10.1515/fca-2019-0074 10.2996/kmj/1414674621 10.1007/s10231-021-01177-4 10.1080/03605302.2021.1998908 10.1016/j.aml.2022.108177 10.1007/s00526-024-02819-0 10.1016/j.dam.2007.05.013 10.1090/tran/8621 10.1007/s00526-024-02670-3 10.1016/j.matpur.2021.01.008 10.1007/s00526-022-02401-6 10.1007/s12220-022-01173-x 10.1016/0362-546X(86)90011-8 10.1016/j.jde.2023.02.049 10.1007/s10474-018-0894-8 10.1016/j.jde.2021.03.036 10.1016/j.indag.2017.09.008 10.4171/rmi/1342 10.1142/S0219199722500328 10.1016/j.anihpc.2015.04.003 10.1007/s00526-013-0600-1 10.1007/s00526-020-1708-8 10.1016/j.jde.2025.01.030 10.1017/prm.2020.75 10.1515/ans-2016-0010 10.1016/j.jmaa.2016.08.048 10.1016/j.na.2022.113022 10.1007/s00208-022-02512-7 10.1016/S0362-546X(97)00530-0 10.1142/S0219199722500572 10.7494/OpMath.2020.40.4.483 10.1017/prm.2023.80 10.3934/dcds.2022126 10.1007/s00208-006-0053-3 10.1214/EJP.v14-604 10.1016/j.bulsci.2011.12.004 10.1016/j.physa.2021.126052 |
| ContentType | Journal Article |
| Copyright | 2025 Elsevier Ltd |
| Copyright_xml | – notice: 2025 Elsevier Ltd |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.na.2025.113796 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Applied Sciences Engineering Mathematics |
| ExternalDocumentID | 10_1016_j_na_2025_113796 S0362546X25000513 |
| GroupedDBID | --K --M -~X .~1 0R~ 123 1B1 1RT 1~. 1~5 29N 4.4 457 4G. 5VS 7-5 71M 8P~ 9JN AABNK AACTN AAEDT AAEDW AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AATTM AAXKI AAXUO ABAOU ABEFU ABFNM ABJNI ABLJU ABMAC ABWVN ABXDB ACDAQ ACGFS ACIWK ACNNM ACRLP ACRPL ADBBV ADEZE ADGUI ADIYS ADMUD ADNMO ADTZH AEBSH AECPX AEIPS AEKER AFJKZ AFTJW AFXIZ AGHFR AGQPQ AGUBO AGYEJ AHHHB AHJVU AIEXJ AIGVJ AIKHN AITUG ALMA_UNASSIGNED_HOLDINGS AMRAJ ANKPU APXCP ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BLXMC BNPGV CS3 EBS EFJIC EJD EO8 EO9 EP2 EP3 FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 GBLVA HMJ HVGLF HZ~ IHE J1W JJJVA KOM M26 M41 MHUIS MO0 MVM N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. Q38 R2- RIG ROL RPZ SDF SDG SDP SES SEW SME SPC SPCBC SSH SST SSW SSZ T5K TN5 WUQ XPP YQT ZMT ~G- AAYWO AAYXX AGCQF AIIUN CITATION EFKBS EFLBG |
| ID | FETCH-LOGICAL-c177t-e67ec284b72f291135b4327e967565bd67047783d27aff543dec0a032805a3513 |
| IEDL.DBID | AIKHN |
| ISSN | 0362-546X |
| IngestDate | Wed Sep 10 05:19:27 EDT 2025 Sat Apr 19 15:56:44 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | 35J62 35B45 35A01 35K59 35P30 35M10 Weak Harnack inequality Elliptic and doubly nonlinear parabolic mixed anisotropic and nonlocal quasilinear equation Pointwise behavior Energy estimates 35B65 35R11 35B51 System of mixed singular equations 35B50 Regularity Semicontinuity Existence De Giorgi theory |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c177t-e67ec284b72f291135b4327e967565bd67047783d27aff543dec0a032805a3513 |
| ParticipantIDs | crossref_primary_10_1016_j_na_2025_113796 elsevier_sciencedirect_doi_10_1016_j_na_2025_113796 |
| PublicationCentury | 2000 |
| PublicationDate | July 2025 2025-07-00 |
| PublicationDateYYYYMMDD | 2025-07-01 |
| PublicationDate_xml | – month: 07 year: 2025 text: July 2025 |
| PublicationDecade | 2020 |
| PublicationTitle | Nonlinear analysis |
| PublicationYear | 2025 |
| Publisher | Elsevier Ltd |
| Publisher_xml | – name: Elsevier Ltd |
| References | Banerjee, Garain, Kinnunen (b8) 2022; 201 Garain, Ukhlov (b44) 2023; 14 Garain, Kinnunen (b39) 2022; 375 Garain (b37) 2024; 281 Giacomoni, Gouasmia, Mokrane (b45) 2022; 116 Brasco, Franzina (b21) 2014; 37 Buccheri, da Silva, de Miranda (b24) 2022; 128 Liao (b50) 2021; 147 Dipierro, Proietti, Valdinoci (b29) 2022; 128 Fragnelli, Mugnai, Papageorgiou (b35) 2016; 16 Pezzo, Ferreira, Rossi (b58) 2019; 22 Lindgren, Lindqvist (b51) 2014; 49 Mezei, Vas (b53) 2019; 157 Adimurthi, Prasad, Tewary (b1) 2023 Salort, Vecchi (b61) 2022; 35 Dwivedi, Tyagi (b31) 2017; 446 Garain, Kim, Kinnunen (b38) 2024; 36 Brasco, Parini (b22) 2016; 9 Banerjee, Garain, Kinnunen (b9) 2023; 25 DiBenedetto (b28) 1993 Foondun (b34) 2009; 14 Arora, Radulescu (b5) 2023 Radulescu, Shang, Zhang (b60) 2024 Gianazza, Vespri (b46) 2006; 1 Biagi, Dipierro, Valdinoci, Vecchi (b13) 2023; 5 Biagi, Vecchi, Dipierro, Valdinoci (b17) 2021; 151 Barlow, Bass, Chen, Kassmann (b10) 2009; 361 Bögelein, Duzaar, Liao, Schätzler (b20) 2023; 39 Nakamura (b55) 2022; 22 Antonini, Cozzi (b4) 2025; 425 Bal (b6) 2013; 2013 Brézis, Oswald (b23) 1986; 10 Castro, Kuusi, Palatucci (b26) 2016; 33 Filippis, Mingione (b33) 2024; 388 Mugnai, Pinamonti, Vecchi (b54) 2020; 59 Biagi, Vecchi (b16) 2024; 63 Amghibech (b3) 2008; 156 Bögelein, Duzaar, Gianazza, Liao, Scheven (b18) 2023 Dipierro, Valdinoci (b30) 2021; 575 Diaz, Saa (b27) 1987; 305 Allegretto, Huang (b2) 1998; 32 Garain, Kinnunen (b41) 2024; 25 Xia (b65) 2012 Bal, Garain, Mandal (b7) 2017; 28 Shang, Zhang (b63) 2023; 33 Castro, Kuusi, Palatucci (b25) 2014; 267 Jaroš (b48) 2015; 143 Kinnunen, Kuusi (b49) 2007; 337 Nezza, Palatucci, Valdinoci (b57) 2012; 136 Garain (b36) 2020; 40 Belloni, Ferone, Kawohl (b11) 2003; 54 Garain, Lindgren (b42) 2023; 62 Malý, Ziemer (b52) 1997; vol. 51 Shang, Zhang (b62) 2022; 42 Farkas, Winkert (b32) 2021; 286 Garain, Kinnunen (b40) 2023; 360 Shang, Zhang (b64) 2023; 133 Biagi, Mugnai, Vecchi (b15) 2024; 26 Bögelein, Duzaar, Liao (b19) 2021; 281 Nakamura (b56) 2023; 62 Heinonen, Kilpeläinen, Martio (b47) 2018 Biagi, Mugnai, Vecchi (b14) 2022; 132 Garain, Ukhlov (b43) 2022; 223 Biagi, Dipierro, Valdinoci, Vecchi (b12) 2022; 47 Prasad (b59) 2024; 63 Castro (10.1016/j.na.2025.113796_b26) 2016; 33 Dipierro (10.1016/j.na.2025.113796_b30) 2021; 575 Adimurthi (10.1016/j.na.2025.113796_b1) 2023 Biagi (10.1016/j.na.2025.113796_b16) 2024; 63 Liao (10.1016/j.na.2025.113796_b50) 2021; 147 Bal (10.1016/j.na.2025.113796_b7) 2017; 28 Biagi (10.1016/j.na.2025.113796_b17) 2021; 151 Heinonen (10.1016/j.na.2025.113796_b47) 2018 Buccheri (10.1016/j.na.2025.113796_b24) 2022; 128 Biagi (10.1016/j.na.2025.113796_b13) 2023; 5 Farkas (10.1016/j.na.2025.113796_b32) 2021; 286 Barlow (10.1016/j.na.2025.113796_b10) 2009; 361 Amghibech (10.1016/j.na.2025.113796_b3) 2008; 156 Nezza (10.1016/j.na.2025.113796_b57) 2012; 136 Diaz (10.1016/j.na.2025.113796_b27) 1987; 305 Radulescu (10.1016/j.na.2025.113796_b60) 2024 Garain (10.1016/j.na.2025.113796_b43) 2022; 223 Garain (10.1016/j.na.2025.113796_b39) 2022; 375 Brasco (10.1016/j.na.2025.113796_b21) 2014; 37 Biagi (10.1016/j.na.2025.113796_b14) 2022; 132 Belloni (10.1016/j.na.2025.113796_b11) 2003; 54 Banerjee (10.1016/j.na.2025.113796_b9) 2023; 25 Arora (10.1016/j.na.2025.113796_b5) 2023 Banerjee (10.1016/j.na.2025.113796_b8) 2022; 201 Biagi (10.1016/j.na.2025.113796_b15) 2024; 26 Nakamura (10.1016/j.na.2025.113796_b56) 2023; 62 Garain (10.1016/j.na.2025.113796_b36) 2020; 40 Garain (10.1016/j.na.2025.113796_b40) 2023; 360 Garain (10.1016/j.na.2025.113796_b41) 2024; 25 Garain (10.1016/j.na.2025.113796_b44) 2023; 14 Shang (10.1016/j.na.2025.113796_b64) 2023; 133 Garain (10.1016/j.na.2025.113796_b38) 2024; 36 Dwivedi (10.1016/j.na.2025.113796_b31) 2017; 446 Shang (10.1016/j.na.2025.113796_b63) 2023; 33 Brézis (10.1016/j.na.2025.113796_b23) 1986; 10 Fragnelli (10.1016/j.na.2025.113796_b35) 2016; 16 Bögelein (10.1016/j.na.2025.113796_b20) 2023; 39 Salort (10.1016/j.na.2025.113796_b61) 2022; 35 Pezzo (10.1016/j.na.2025.113796_b58) 2019; 22 Bal (10.1016/j.na.2025.113796_b6) 2013; 2013 Castro (10.1016/j.na.2025.113796_b25) 2014; 267 Malý (10.1016/j.na.2025.113796_b52) 1997; vol. 51 Lindgren (10.1016/j.na.2025.113796_b51) 2014; 49 Gianazza (10.1016/j.na.2025.113796_b46) 2006; 1 Garain (10.1016/j.na.2025.113796_b42) 2023; 62 Filippis (10.1016/j.na.2025.113796_b33) 2024; 388 Kinnunen (10.1016/j.na.2025.113796_b49) 2007; 337 Jaroš (10.1016/j.na.2025.113796_b48) 2015; 143 Brasco (10.1016/j.na.2025.113796_b22) 2016; 9 Foondun (10.1016/j.na.2025.113796_b34) 2009; 14 Shang (10.1016/j.na.2025.113796_b62) 2022; 42 Nakamura (10.1016/j.na.2025.113796_b55) 2022; 22 Allegretto (10.1016/j.na.2025.113796_b2) 1998; 32 Antonini (10.1016/j.na.2025.113796_b4) 2025; 425 Giacomoni (10.1016/j.na.2025.113796_b45) 2022; 116 Prasad (10.1016/j.na.2025.113796_b59) 2024; 63 Xia (10.1016/j.na.2025.113796_b65) 2012 Dipierro (10.1016/j.na.2025.113796_b29) 2022; 128 Mezei (10.1016/j.na.2025.113796_b53) 2019; 157 Biagi (10.1016/j.na.2025.113796_b12) 2022; 47 Bögelein (10.1016/j.na.2025.113796_b18) 2023 DiBenedetto (10.1016/j.na.2025.113796_b28) 1993 Bögelein (10.1016/j.na.2025.113796_b19) 2021; 281 Mugnai (10.1016/j.na.2025.113796_b54) 2020; 59 Garain (10.1016/j.na.2025.113796_b37) 2024; 281 |
| References_xml | – volume: 136 start-page: 521 year: 2012 end-page: 573 ident: b57 article-title: Hitchhiker’s guide to the fractional Sobolev spaces publication-title: Bull. Sci. Math. – volume: 267 start-page: 1807 year: 2014 end-page: 1836 ident: b25 article-title: Nonlocal Harnack inequalities publication-title: J. Funct. Anal. – year: 2012 ident: b65 article-title: On a class of anisotropic problems publication-title: Dissertation zur Erlangung des Doktorgrades der Fakultät Mathematik und Physik der Albert-Ludwigs-Universität tFreiburg – year: 2023 ident: b18 article-title: Hölder continuity of the gradient of solutions to doubly non-linear parabolic equations – volume: 1 start-page: 271 year: 2006 end-page: 284 ident: b46 article-title: A Harnack inequality for solutions of double nonlinearity parabolic equations publication-title: J. Appl. Funct. Anal. – volume: 33 start-page: 24 year: 2023 ident: b63 article-title: Harnack inequality for mixed local and nonlocal parabolic publication-title: J. Geom. Anal. – volume: 63 start-page: 19 year: 2024 ident: b59 article-title: On the weak Harnack estimate for nonlocal equations publication-title: Calc. Var. Partial Differential Equations – volume: 281 start-page: 58 year: 2021 ident: b19 article-title: On the Hölder regularity of signed solutions to a doubly nonlinear equation publication-title: J. Funct. Anal. – volume: 575 start-page: 20 year: 2021 ident: b30 article-title: Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes publication-title: Phys. A – volume: 201 start-page: 1717 year: 2022 end-page: 1751 ident: b8 article-title: Some local properties of subsolution and supersolutions for a doubly nonlinear nonlocal publication-title: Ann. Mat. Pura Appl. (4) – volume: 388 start-page: 261 year: 2024 end-page: 328 ident: b33 article-title: Gradient regularity in mixed local and nonlocal problems publication-title: Math. Ann. – volume: 39 start-page: 1005 year: 2023 end-page: 1037 ident: b20 article-title: On the Hölder regularity of signed solutions to a doubly nonlinear equation. II publication-title: Rev. Mat. Iberoam. – year: 1993 ident: b28 article-title: Degenerate parabolic equations publication-title: Universitext – volume: 10 start-page: 55 year: 1986 end-page: 64 ident: b23 article-title: Remarks sublinear elliptic equations publication-title: Nonlinear Anal. Theory Methods Appl. – volume: 40 start-page: 483 year: 2020 end-page: 494 ident: b36 article-title: Properties of solutions to some weighted publication-title: Opusc. Math. – volume: 62 start-page: 36 year: 2023 ident: b42 article-title: Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations publication-title: Calc. Var. Partial. Differ. Equ. – year: 2018 ident: b47 article-title: Nonlinear potential theory of degenerate elliptic equations. With a new preface, corrigenda, and epilogue consisting of other new material publication-title: Dover Books Math – volume: 62 start-page: 45 year: 2023 ident: b56 article-title: Harnack’s estimate for a mixed local-nonlocal doubly nonlinear parabolic equation publication-title: Calc. Var. Partial Differential Equations – volume: 151 start-page: 1611 year: 2021 end-page: 1641 ident: b17 article-title: Semilinear elliptic equations involving mixed local and nonlocal operators publication-title: Proc. R. Soc. Edinb. Sect. A, Math. – volume: 33 start-page: 1279 year: 2016 end-page: 1299 ident: b26 article-title: Local behavior of fractional publication-title: Ann. Inst. Henri Poincaré, Anal. Non Linéaire – year: 2024 ident: b60 article-title: Harnack inequality for doubly nonlinear mixed local and nonlocal parabolic equations – volume: 25 start-page: 495 year: 2024 end-page: 541 ident: b41 article-title: On the regularity theory for mixed local and nonlocal quasilinear parabolic equations publication-title: Ann. Sc. Norm. Super. Pisa, Cl. Sci. ( 5) – volume: 42 start-page: 5817 year: 2022 end-page: 5837 ident: b62 article-title: Hölder regularity for mixed local and nonlocal publication-title: Discrete Contin. Dyn. Syst. – volume: 337 start-page: 705 year: 2007 end-page: 728 ident: b49 article-title: Local behaviour of solutions to doubly nonlinear parabolic equations publication-title: Math. Ann. – volume: 361 start-page: 1963 year: 2009 end-page: 1999 ident: b10 article-title: Non-local Dirichlet forms and symmetric jump processes publication-title: Trans. Amer. Math. Soc. – volume: 9 start-page: 323 year: 2016 end-page: 355 ident: b22 article-title: The second eigenvalue of the fractional publication-title: Adv. Calc. Var. – volume: 22 start-page: 38 year: 2022 ident: b55 article-title: Local boundedness of a mixed local-nonlocal doubly nonlinear equation publication-title: J. Evol. Equ. – volume: 156 start-page: 1 year: 2008 end-page: 10 ident: b3 article-title: On the discrete version of Picone’s identity publication-title: Discrete Appl. Math. – volume: 360 start-page: 373 year: 2023 end-page: 406 ident: b40 article-title: Weak Harnack inequality for a mixed local and nonlocal parabolic equation publication-title: J. Differ. Equ. – volume: vol. 51 year: 1997 ident: b52 article-title: Fine regularity of solutions of elliptic partial differential equations publication-title: Math. Surv. Monogr. – volume: 147 start-page: 179 year: 2021 end-page: 204 ident: b50 article-title: Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior publication-title: J. Math. Pures Appl. (9) – volume: 28 start-page: 1258 year: 2017 end-page: 1264 ident: b7 article-title: Some qualitative properties of Finsler publication-title: Indag. Math. New Ser. – volume: 128 start-page: 149 year: 2022 end-page: 181 ident: b24 article-title: A system of local/nonlocal publication-title: Asymptot. Anal. – volume: 36 start-page: 697 year: 2024 end-page: 715 ident: b38 article-title: On the regularity theory for mixed anisotropic and nonlocal publication-title: Forum Math. – volume: 223 start-page: 35 year: 2022 ident: b43 article-title: Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems publication-title: Nonlinear Anal. Theory Methods Appl. Ser. A, Theory Methods – volume: 425 start-page: 342 year: 2025 end-page: 382 ident: b4 article-title: Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type, 2023 publication-title: J. Differential Equations – volume: 54 start-page: 771 year: 2003 end-page: 783 ident: b11 article-title: Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators publication-title: Z. Angew. Math. Phys. – volume: 25 start-page: 23 year: 2023 ident: b9 article-title: Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic publication-title: Commun. Contemp. Math. – volume: 305 start-page: 521 year: 1987 end-page: 524 ident: b27 article-title: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. (Existence and uniqueness of solutions of some quasilinear elliptic equations) publication-title: C. R. Acad. Sci. Paris, Sér. I – volume: 37 start-page: 769 year: 2014 end-page: 799 ident: b21 article-title: Convexity properties of Dirichlet integrals and Picone-type inequalities publication-title: Kodai Math. J. – volume: 59 start-page: 25 year: 2020 ident: b54 article-title: Towards a Brezis-Oswald-type result for fractional problems with Robin boundary conditions publication-title: Calc. Var. Partial Differential Equations – volume: 132 start-page: 9 year: 2022 ident: b14 article-title: Necessary condition in a Brezis-Oswald-type problem for mixed local and nonlocal operators publication-title: Appl. Math. Lett. – volume: 32 start-page: 819 year: 1998 end-page: 830 ident: b2 article-title: A Picone’s identity for the publication-title: Nonlinear Anal. Theory Methods Appl. – volume: 157 start-page: 39 year: 2019 end-page: 53 ident: b53 article-title: Existence results for some Dirichlet problems involving Finsler-Laplacian operator publication-title: Acta Math. Hung. – volume: 63 start-page: 45 year: 2024 ident: b16 article-title: Multiplicity of positive solutions for mixed local-nonlocal singular critical problems publication-title: Calc. Var. Partial Differential Equations – volume: 49 start-page: 795 year: 2014 end-page: 826 ident: b51 article-title: Fractional eigenvalues publication-title: Calc. Var. Partial Differential Equations – volume: 2013 start-page: 6 year: 2013 ident: b6 article-title: Generalized Picone’s identity and its applications publication-title: Electron. J. Differential Equations – volume: 375 start-page: 5393 year: 2022 end-page: 5423 ident: b39 article-title: On the regularity theory for mixed local and nonlocal quasilinear elliptic equations publication-title: Trans. Am. Math. Soc. – volume: 35 start-page: 795 year: 2022 end-page: 818 ident: b61 article-title: On the mixed local-nonlocal Hénon equation publication-title: Differ. Integral Equ. Appl. – volume: 446 start-page: 456 year: 2017 end-page: 469 ident: b31 article-title: Some qualitative questions on the equation publication-title: J. Math. Anal. Appl. – volume: 22 start-page: 1414 year: 2019 end-page: 1436 ident: b58 article-title: Eigenvalues for a combination between local and nonlocal publication-title: Fract. Calc. Appl. Anal. – volume: 14 start-page: 314 year: 2009 end-page: 340 ident: b34 article-title: Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part publication-title: Electron. J. Probab. – year: 2023 ident: b1 article-title: Gradient regularity for mixed local-nonlocal quasilinear parabolic equations – volume: 26 start-page: 28 year: 2024 ident: b15 article-title: A Brezis-Oswald approach for mixed local and nonlocal operators publication-title: Commun. Contemp. Math. – volume: 14 start-page: 217 year: 2023 end-page: 232 ident: b44 article-title: Mixed anisotropic and nonlocal Dirichlet publication-title: Riv. Mat. Univ. Parma ( N. S.) – volume: 281 start-page: 633 year: 2024 end-page: 645 ident: b37 article-title: Mixed anisotropic and nonlocal Sobolev type inequalities with extremal publication-title: J. Math. Sci. N. Y. – volume: 133 start-page: 1 year: 2023 end-page: 12 ident: b64 article-title: A strong maximum principle for mixed local and nonlocal publication-title: Asymptot. Anal. – volume: 47 start-page: 585 year: 2022 end-page: 629 ident: b12 article-title: Mixed local and nonlocal elliptic operators: regularity and maximum principles publication-title: Commun. Partial. Differ. Equations – year: 2023 ident: b5 article-title: Combined effects in mixed local-nonlocal stationary problems publication-title: Proc. R. Soc. Edinb.: Sect. A Math. – volume: 128 start-page: 571 year: 2022 end-page: 594 ident: b29 article-title: Linear theory for a mixed operator with Neumann conditions publication-title: Asymptot. Anal. – volume: 16 start-page: 603 year: 2016 end-page: 622 ident: b35 article-title: The Brezis-Oswald result for quasilinear Robin problems publication-title: Adv. Nonlinear Stud. – volume: 116 start-page: 21 year: 2022 ident: b45 article-title: Discrete Picone inequalities and applications to non local and non homogenenous operators publication-title: Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM – volume: 5 start-page: 25 year: 2023 ident: b13 article-title: A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators publication-title: Math. Eng. ( Springfield) – volume: 286 start-page: 455 year: 2021 end-page: 473 ident: b32 article-title: An existence result for singular Finsler double phase problems publication-title: J. Differential Equations – volume: 143 start-page: 1137 year: 2015 end-page: 1144 ident: b48 article-title: Caccioppoli estimates through an anisotropic Picone’s identity publication-title: Proc. Amer. Math. Soc. – year: 1993 ident: 10.1016/j.na.2025.113796_b28 article-title: Degenerate parabolic equations – volume: 9 start-page: 323 issue: 4 year: 2016 ident: 10.1016/j.na.2025.113796_b22 article-title: The second eigenvalue of the fractional p-Laplacian publication-title: Adv. Calc. Var. doi: 10.1515/acv-2015-0007 – volume: 143 start-page: 1137 issue: 3 year: 2015 ident: 10.1016/j.na.2025.113796_b48 article-title: Caccioppoli estimates through an anisotropic Picone’s identity publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-2014-12294-5 – volume: 54 start-page: 771 issue: 5 year: 2003 ident: 10.1016/j.na.2025.113796_b11 article-title: Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators publication-title: Z. Angew. Math. Phys. doi: 10.1007/s00033-003-3209-y – volume: 281 start-page: 58 issue: 9 year: 2021 ident: 10.1016/j.na.2025.113796_b19 article-title: On the Hölder regularity of signed solutions to a doubly nonlinear equation publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2021.109173 – volume: 22 start-page: 38 issue: 3 year: 2022 ident: 10.1016/j.na.2025.113796_b55 article-title: Local boundedness of a mixed local-nonlocal doubly nonlinear equation publication-title: J. Evol. Equ. doi: 10.1007/s00028-022-00832-4 – volume: 267 start-page: 1807 issue: 6 year: 2014 ident: 10.1016/j.na.2025.113796_b25 article-title: Nonlocal Harnack inequalities publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2014.05.023 – volume: 36 start-page: 697 issue: 3 year: 2024 ident: 10.1016/j.na.2025.113796_b38 article-title: On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems publication-title: Forum Math. doi: 10.1515/forum-2023-0151 – volume: 62 start-page: 45 issue: 2 year: 2023 ident: 10.1016/j.na.2025.113796_b56 article-title: Harnack’s estimate for a mixed local-nonlocal doubly nonlinear parabolic equation publication-title: Calc. Var. Partial Differential Equations doi: 10.1007/s00526-022-02378-2 – volume: 25 start-page: 495 issue: 1 year: 2024 ident: 10.1016/j.na.2025.113796_b41 article-title: On the regularity theory for mixed local and nonlocal quasilinear parabolic equations publication-title: Ann. Sc. Norm. Super. Pisa, Cl. Sci. ( 5) – volume: 1 start-page: 271 issue: 3 year: 2006 ident: 10.1016/j.na.2025.113796_b46 article-title: A Harnack inequality for solutions of double nonlinearity parabolic equations publication-title: J. Appl. Funct. Anal. – volume: 14 start-page: 217 issue: 1 year: 2023 ident: 10.1016/j.na.2025.113796_b44 article-title: Mixed anisotropic and nonlocal Dirichlet (p,q)-eigenvalue problem publication-title: Riv. Mat. Univ. Parma ( N. S.) – volume: 361 start-page: 1963 issue: 4 year: 2009 ident: 10.1016/j.na.2025.113796_b10 article-title: Non-local Dirichlet forms and symmetric jump processes publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-08-04544-3 – volume: 281 start-page: 633 issue: 5 year: 2024 ident: 10.1016/j.na.2025.113796_b37 article-title: Mixed anisotropic and nonlocal Sobolev type inequalities with extremal publication-title: J. Math. Sci. N. Y. doi: 10.1007/s10958-024-07141-w – volume: 116 start-page: 21 issue: 3 year: 2022 ident: 10.1016/j.na.2025.113796_b45 article-title: Discrete Picone inequalities and applications to non local and non homogenenous operators publication-title: Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM – volume: 22 start-page: 1414 issue: 5 year: 2019 ident: 10.1016/j.na.2025.113796_b58 article-title: Eigenvalues for a combination between local and nonlocal p-Laplacians publication-title: Fract. Calc. Appl. Anal. doi: 10.1515/fca-2019-0074 – volume: 37 start-page: 769 issue: 3 year: 2014 ident: 10.1016/j.na.2025.113796_b21 article-title: Convexity properties of Dirichlet integrals and Picone-type inequalities publication-title: Kodai Math. J. doi: 10.2996/kmj/1414674621 – volume: 201 start-page: 1717 issue: 4 year: 2022 ident: 10.1016/j.na.2025.113796_b8 article-title: Some local properties of subsolution and supersolutions for a doubly nonlinear nonlocal p-Laplace equation publication-title: Ann. Mat. Pura Appl. (4) doi: 10.1007/s10231-021-01177-4 – volume: 47 start-page: 585 issue: 3 year: 2022 ident: 10.1016/j.na.2025.113796_b12 article-title: Mixed local and nonlocal elliptic operators: regularity and maximum principles publication-title: Commun. Partial. Differ. Equations doi: 10.1080/03605302.2021.1998908 – volume: 132 start-page: 9 year: 2022 ident: 10.1016/j.na.2025.113796_b14 article-title: Necessary condition in a Brezis-Oswald-type problem for mixed local and nonlocal operators publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2022.108177 – volume: 63 start-page: 45 issue: 9 year: 2024 ident: 10.1016/j.na.2025.113796_b16 article-title: Multiplicity of positive solutions for mixed local-nonlocal singular critical problems publication-title: Calc. Var. Partial Differential Equations doi: 10.1007/s00526-024-02819-0 – volume: 133 start-page: 1 issue: 1–2 year: 2023 ident: 10.1016/j.na.2025.113796_b64 article-title: A strong maximum principle for mixed local and nonlocal p-Laplace equations publication-title: Asymptot. Anal. – volume: 156 start-page: 1 issue: 1 year: 2008 ident: 10.1016/j.na.2025.113796_b3 article-title: On the discrete version of Picone’s identity publication-title: Discrete Appl. Math. doi: 10.1016/j.dam.2007.05.013 – volume: 375 start-page: 5393 issue: 8 year: 2022 ident: 10.1016/j.na.2025.113796_b39 article-title: On the regularity theory for mixed local and nonlocal quasilinear elliptic equations publication-title: Trans. Am. Math. Soc. doi: 10.1090/tran/8621 – volume: 63 start-page: 19 issue: 3 year: 2024 ident: 10.1016/j.na.2025.113796_b59 article-title: On the weak Harnack estimate for nonlocal equations publication-title: Calc. Var. Partial Differential Equations doi: 10.1007/s00526-024-02670-3 – volume: 147 start-page: 179 year: 2021 ident: 10.1016/j.na.2025.113796_b50 article-title: Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior publication-title: J. Math. Pures Appl. (9) doi: 10.1016/j.matpur.2021.01.008 – volume: 128 start-page: 149 issue: 2 year: 2022 ident: 10.1016/j.na.2025.113796_b24 article-title: A system of local/nonlocal p-laplacians: the eigenvalue problem and its asymptotic limit as p→∞ publication-title: Asymptot. Anal. – volume: 62 start-page: 36 issue: 2 year: 2023 ident: 10.1016/j.na.2025.113796_b42 article-title: Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations publication-title: Calc. Var. Partial. Differ. Equ. doi: 10.1007/s00526-022-02401-6 – volume: 33 start-page: 24 issue: 4 year: 2023 ident: 10.1016/j.na.2025.113796_b63 article-title: Harnack inequality for mixed local and nonlocal parabolic p-Laplace equations publication-title: J. Geom. Anal. doi: 10.1007/s12220-022-01173-x – volume: 10 start-page: 55 year: 1986 ident: 10.1016/j.na.2025.113796_b23 article-title: Remarks sublinear elliptic equations publication-title: Nonlinear Anal. Theory Methods Appl. doi: 10.1016/0362-546X(86)90011-8 – volume: 360 start-page: 373 year: 2023 ident: 10.1016/j.na.2025.113796_b40 article-title: Weak Harnack inequality for a mixed local and nonlocal parabolic equation publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2023.02.049 – year: 2024 ident: 10.1016/j.na.2025.113796_b60 – year: 2012 ident: 10.1016/j.na.2025.113796_b65 article-title: On a class of anisotropic problems – volume: 157 start-page: 39 issue: 1 year: 2019 ident: 10.1016/j.na.2025.113796_b53 article-title: Existence results for some Dirichlet problems involving Finsler-Laplacian operator publication-title: Acta Math. Hung. doi: 10.1007/s10474-018-0894-8 – volume: 286 start-page: 455 year: 2021 ident: 10.1016/j.na.2025.113796_b32 article-title: An existence result for singular Finsler double phase problems publication-title: J. Differential Equations doi: 10.1016/j.jde.2021.03.036 – volume: 28 start-page: 1258 issue: 6 year: 2017 ident: 10.1016/j.na.2025.113796_b7 article-title: Some qualitative properties of Finsler p-Laplacian publication-title: Indag. Math. New Ser. doi: 10.1016/j.indag.2017.09.008 – volume: 128 start-page: 571 issue: 4 year: 2022 ident: 10.1016/j.na.2025.113796_b29 article-title: Linear theory for a mixed operator with Neumann conditions publication-title: Asymptot. Anal. – volume: 39 start-page: 1005 issue: 3 year: 2023 ident: 10.1016/j.na.2025.113796_b20 article-title: On the Hölder regularity of signed solutions to a doubly nonlinear equation. II publication-title: Rev. Mat. Iberoam. doi: 10.4171/rmi/1342 – volume: 5 start-page: 25 issue: 1 year: 2023 ident: 10.1016/j.na.2025.113796_b13 article-title: A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators publication-title: Math. Eng. ( Springfield) – volume: 25 start-page: 23 issue: 8 year: 2023 ident: 10.1016/j.na.2025.113796_b9 article-title: Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p-Laplace equations publication-title: Commun. Contemp. Math. doi: 10.1142/S0219199722500328 – volume: 33 start-page: 1279 issue: 5 year: 2016 ident: 10.1016/j.na.2025.113796_b26 article-title: Local behavior of fractional p-minimizers publication-title: Ann. Inst. Henri Poincaré, Anal. Non Linéaire doi: 10.1016/j.anihpc.2015.04.003 – year: 2018 ident: 10.1016/j.na.2025.113796_b47 article-title: Nonlinear potential theory of degenerate elliptic equations. With a new preface, corrigenda, and epilogue consisting of other new material – volume: 49 start-page: 795 issue: 1–2 year: 2014 ident: 10.1016/j.na.2025.113796_b51 article-title: Fractional eigenvalues publication-title: Calc. Var. Partial Differential Equations doi: 10.1007/s00526-013-0600-1 – volume: 59 start-page: 25 issue: 2 year: 2020 ident: 10.1016/j.na.2025.113796_b54 article-title: Towards a Brezis-Oswald-type result for fractional problems with Robin boundary conditions publication-title: Calc. Var. Partial Differential Equations doi: 10.1007/s00526-020-1708-8 – year: 2023 ident: 10.1016/j.na.2025.113796_b1 – volume: 425 start-page: 342 year: 2025 ident: 10.1016/j.na.2025.113796_b4 article-title: Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type, 2023 publication-title: J. Differential Equations doi: 10.1016/j.jde.2025.01.030 – volume: 151 start-page: 1611 issue: 5 year: 2021 ident: 10.1016/j.na.2025.113796_b17 article-title: Semilinear elliptic equations involving mixed local and nonlocal operators publication-title: Proc. R. Soc. Edinb. Sect. A, Math. doi: 10.1017/prm.2020.75 – year: 2023 ident: 10.1016/j.na.2025.113796_b18 – volume: 16 start-page: 603 issue: 3 year: 2016 ident: 10.1016/j.na.2025.113796_b35 article-title: The Brezis-Oswald result for quasilinear Robin problems publication-title: Adv. Nonlinear Stud. doi: 10.1515/ans-2016-0010 – volume: 446 start-page: 456 issue: 1 year: 2017 ident: 10.1016/j.na.2025.113796_b31 article-title: Some qualitative questions on the equation −div(a(x,u,u))=f(x,u) publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2016.08.048 – volume: 2013 start-page: 6 year: 2013 ident: 10.1016/j.na.2025.113796_b6 article-title: Generalized Picone’s identity and its applications publication-title: Electron. J. Differential Equations – volume: 305 start-page: 521 year: 1987 ident: 10.1016/j.na.2025.113796_b27 article-title: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. (Existence and uniqueness of solutions of some quasilinear elliptic equations) publication-title: C. R. Acad. Sci. Paris, Sér. I – volume: 223 start-page: 35 year: 2022 ident: 10.1016/j.na.2025.113796_b43 article-title: Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems publication-title: Nonlinear Anal. Theory Methods Appl. Ser. A, Theory Methods doi: 10.1016/j.na.2022.113022 – volume: 388 start-page: 261 issue: 1 year: 2024 ident: 10.1016/j.na.2025.113796_b33 article-title: Gradient regularity in mixed local and nonlocal problems publication-title: Math. Ann. doi: 10.1007/s00208-022-02512-7 – volume: 32 start-page: 819 issue: 7 year: 1998 ident: 10.1016/j.na.2025.113796_b2 article-title: A Picone’s identity for the p-Laplacian and applications publication-title: Nonlinear Anal. Theory Methods Appl. doi: 10.1016/S0362-546X(97)00530-0 – volume: 26 start-page: 28 issue: 2 year: 2024 ident: 10.1016/j.na.2025.113796_b15 article-title: A Brezis-Oswald approach for mixed local and nonlocal operators publication-title: Commun. Contemp. Math. doi: 10.1142/S0219199722500572 – volume: vol. 51 year: 1997 ident: 10.1016/j.na.2025.113796_b52 article-title: Fine regularity of solutions of elliptic partial differential equations – volume: 35 start-page: 795 issue: 11–12 year: 2022 ident: 10.1016/j.na.2025.113796_b61 article-title: On the mixed local-nonlocal Hénon equation publication-title: Differ. Integral Equ. Appl. – volume: 40 start-page: 483 issue: 4 year: 2020 ident: 10.1016/j.na.2025.113796_b36 article-title: Properties of solutions to some weighted p-Laplacian equation publication-title: Opusc. Math. doi: 10.7494/OpMath.2020.40.4.483 – year: 2023 ident: 10.1016/j.na.2025.113796_b5 article-title: Combined effects in mixed local-nonlocal stationary problems publication-title: Proc. R. Soc. Edinb.: Sect. A Math. doi: 10.1017/prm.2023.80 – volume: 42 start-page: 5817 issue: 12 year: 2022 ident: 10.1016/j.na.2025.113796_b62 article-title: Hölder regularity for mixed local and nonlocal p-Laplace parabolic equations publication-title: Discrete Contin. Dyn. Syst. doi: 10.3934/dcds.2022126 – volume: 337 start-page: 705 issue: 3 year: 2007 ident: 10.1016/j.na.2025.113796_b49 article-title: Local behaviour of solutions to doubly nonlinear parabolic equations publication-title: Math. Ann. doi: 10.1007/s00208-006-0053-3 – volume: 14 start-page: 314 year: 2009 ident: 10.1016/j.na.2025.113796_b34 article-title: Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part publication-title: Electron. J. Probab. doi: 10.1214/EJP.v14-604 – volume: 136 start-page: 521 issue: 5 year: 2012 ident: 10.1016/j.na.2025.113796_b57 article-title: Hitchhiker’s guide to the fractional Sobolev spaces publication-title: Bull. Sci. Math. doi: 10.1016/j.bulsci.2011.12.004 – volume: 575 start-page: 20 year: 2021 ident: 10.1016/j.na.2025.113796_b30 article-title: Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes publication-title: Phys. A doi: 10.1016/j.physa.2021.126052 |
| SSID | ssj0001736 |
| Score | 2.4458697 |
| Snippet | This article is divided into two parts. In the first part, we examine the Brezis–Oswald problem involving a mixed anisotropic and nonlocal p-Laplace operator.... |
| SourceID | crossref elsevier |
| SourceType | Index Database Publisher |
| StartPage | 113796 |
| SubjectTerms | De Giorgi theory Elliptic and doubly nonlinear parabolic mixed anisotropic and nonlocal quasilinear equation Energy estimates Existence Pointwise behavior Regularity Semicontinuity System of mixed singular equations Weak Harnack inequality |
| Title | Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations |
| URI | https://dx.doi.org/10.1016/j.na.2025.113796 |
| Volume | 256 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1NT9wwEB3BcimHllIQ9AP50AuHsIkd29kjQkULFVwo0t4i23Gk8JEsm6DCpT-F34ondlZUggtHO2M7yjhvnpPxM8BPrrkRIuURzQyLUqZlpDiLI01FqTJjDO9zc87OxfQyPZ3x2QocDXthMK0yYL_H9B6tQ804PM3xvKrGF4i9PBUzynsiwlZhjbKJ4CNYOzz5PT1fAnIimRhUpLBB-Fvp07xqFB-iHM82kajc_1p0ehFxjjfgY6CK5NDfzWdYsfUmfAq0kYSXst2E9Reagq50thRibb_A00Vza4nfOdlLfBNVF1iuu6Fijp_jF6irSpqS_LXqmiznI-kacls9uOFUXbVN50wr03dRN3UfBrGvtkKuqhYExT0dBHmLornXN4-9ob-KKuMaZYiJvfMC4-0WXB7_-nM0jcKRDJFJpOwiK6Q1LqJpSUvqcJJxnTIq7cStOwTXhZBxKmXGCipVWfKUFdbECjX7Yq6Yc9I2jNy4dgcIs4ko6CSzLNOpo0VKOWZCaaILG2ua8V3YH1yRz73yRj6kpF3ltcrRbbl32y6wwVf5f7Mnd4HhzVZf39XqG3zAkk_Z_Q6jbnFvfzhi0uk9WD34l-yF6fcM6QXknQ |
| linkProvider | Elsevier |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LT9wwEB5ROFAO5VlBS8EHLhzCZv0MxwoVbYHlAkh7i2zHkdKWZNkEAZf-lP7WeuKEhwQXjrbHdpRxZr4knz8D7AkjrJRcRDSxLOLMqEgLFkeGylwn1lrRcnPG53J0xU8mYjIHR_1eGKRVdrE_xPQ2Wnc1g-5uDqZFMbjA2Cu4nFDRAhH2ARa4YAp5fQd_n3geQ8VkryGF5t2_ykDyKlF6iAo82UShbv9ruelZvjlegU8dUCTfw7Wswpwr12C5A42keyTrNVh6pijoS-NHGdZ6Hf5dVNeOhH2TrcA30WWG5bLpK6b4MX6Gqqqkysmd07_J42okTUWui3s_nS6Lumq8aWHbIcqqbJMgjlUXiFT1jKC0pw9AwSKrbs2fh9YwtKLGuEERYuJugrx4vQFXxz8uj0ZRdyBDZIdKNZGTylmfz4yiOfVRkgnDGVXu0L91SGEyqWKuVMIyqnSeC84yZ2ONin2x0My76DPM-3ndJhDmhjKjh4ljieEeFGntcQmlQ5O52NBEbMF-74p0GnQ30p6Q9istdYpuS4PbtoD1vkpfrJ3Up4U3e315V69dWBxdjs_Ss5_np1_hI7YE8u42zDezW_fNQ5TG7LRL8D_Z4eVo |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Some+qualitative+and+quantitative+properties+of+weak+solutions+to+mixed+anisotropic+and+nonlocal+quasilinear+elliptic+and+doubly+nonlinear+parabolic+equations&rft.jtitle=Nonlinear+analysis&rft.au=Garain%2C+Prashanta&rft.date=2025-07-01&rft.issn=0362-546X&rft.volume=256&rft.spage=113796&rft_id=info:doi/10.1016%2Fj.na.2025.113796&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_na_2025_113796 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0362-546X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0362-546X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0362-546X&client=summon |