American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2022008
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips

 1 Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam 2 Vietnam National University, Ho Chi Minh City, Vietnam 3 Faculty of Mathematics and Applications, Saigon University, 273 An Duong Vuong St., Ward 3, Dist. 5, Ho Chi Minh City, Vietnam

* Corresponding author

Received  September 2021 Revised  November 2021 Early access December 2021

By means of the method of moving planes, we study the monotonicity of positive solutions to degenerate quasilinear elliptic systems in half-spaces. We also prove the symmetry of positive solutions to the systems in strips by using similar arguments. Our work extends the main results obtained in [16,20] to the system, in which substantial differences with the single cases are presented.

Citation: Phuong Le, Hoang-Hung Vo. Monotonicity and symmetry of positive solutions to degenerate quasilinear elliptic systems in half-spaces and strips. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2022008
References:
 [1] A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl., 58 (1962), 303-315.  doi: 10.1007/BF02413056.  Google Scholar [2] H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Commun. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.   Google Scholar [5] M. F. Bidaut-Véron, R. Borghol and L. Véron, Boundary Harnack inequality and a priori estimates of singular solutions of quasilinear elliptic equations, Calc. Var. Partial Differ Equ, 27 (2006), 159-177.  doi: 10.1007/s00526-006-0003-7.  Google Scholar [6] Z. Chen, C. S. Lin and W. Zou, Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105.  doi: 10.1016/j.jfa.2013.08.021.  Google Scholar [7] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of $p$-Laplace equations, $1 < p < 2$, via the moving plane method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 689-707.   Google Scholar [8] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differ. Equ., 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.  Google Scholar [9] L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of $m$-Laplace equations, Calc. Var. Partial Differ. Equ., 25 (2006), 139-159.  doi: 10.1007/s00526-005-0337-6.  Google Scholar [10] E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254.  doi: 10.1007/s00208-008-0226-3.  Google Scholar [11] D. G. de Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differ. Equ. Appl., 1 (1994), 119-123.  doi: 10.1007/BF01193947.  Google Scholar [12] E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar [13] A. Farina, L. Montoro, G. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1-22.  doi: 10.1016/j.anihpc.2013.09.005.  Google Scholar [14] A. Farina, L. Montoro and B. Sciunzi, Monotonicity and one-dimensional symmetry for solutions of $-\Delta_pu = f(u)$ in half-spaces, Calc. Var. Partial Differ. Equ., 43 (2012), 123-145.  doi: 10.1007/s00526-011-0405-z.  Google Scholar [15] A. Farina, L. Montoro and B. Sciunzi, Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces, Math. Ann., 357 (2013), 855-893.  doi: 10.1007/s00208-013-0919-0.  Google Scholar [16] A. Farina, L. Montoro and B. Sciunzi, Monotonicity in half-space of positive solutions to $-\Delta_pu = f(u)$ in the case $p>2$, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (2017), 1207-1229.   Google Scholar [17] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal., 195 (2010), 1025-1058.  doi: 10.1007/s00205-009-0227-8.  Google Scholar [18] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar [19] T. Kilpeläinen, H. Shahgholian and X. Zhong, Growth estimates through scaling for quasilinear partial differential equations, Ann. Acad. Sci. Fenn. Math., 32 (2007), 595-599.   Google Scholar [20] L. Montoro, G. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differ. Equ., 20 (2015), 717-740.   Google Scholar [21] P. Pucci and J. Serrin, The maximum principle, vol. 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007.  Google Scholar [22] B. Sciunzi, Classification of positive $\mathcal{D}^{1, p}(\mathbb{R}^N)$-solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, Adv. Math., 291 (2016), 12-23.  doi: 10.1016/j.aim.2015.12.028.  Google Scholar [23] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar [24] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar [25] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differ. Equ., 42 (1981), 400-413.  doi: 10.1016/0022-0396(81)90113-3.  Google Scholar [26] J. Vétois, A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations, J. Differ. Equ., 260 (2016), 149-161.  doi: 10.1016/j.jde.2015.08.041.  Google Scholar

show all references

References:
 [1] A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl., 58 (1962), 303-315.  doi: 10.1007/BF02413056.  Google Scholar [2] H. Berestycki, L. A. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Commun. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.  Google Scholar [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94.   Google Scholar [5] M. F. Bidaut-Véron, R. Borghol and L. Véron, Boundary Harnack inequality and a priori estimates of singular solutions of quasilinear elliptic equations, Calc. Var. Partial Differ Equ, 27 (2006), 159-177.  doi: 10.1007/s00526-006-0003-7.  Google Scholar [6] Z. Chen, C. S. Lin and W. Zou, Monotonicity and nonexistence results to cooperative systems in the half space, J. Funct. Anal., 266 (2014), 1088-1105.  doi: 10.1016/j.jfa.2013.08.021.  Google Scholar [7] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of $p$-Laplace equations, $1 < p < 2$, via the moving plane method, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 689-707.   Google Scholar [8] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differ. Equ., 206 (2004), 483-515.  doi: 10.1016/j.jde.2004.05.012.  Google Scholar [9] L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of $m$-Laplace equations, Calc. Var. Partial Differ. Equ., 25 (2006), 139-159.  doi: 10.1007/s00526-005-0337-6.  Google Scholar [10] E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254.  doi: 10.1007/s00208-008-0226-3.  Google Scholar [11] D. G. de Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differ. Equ. Appl., 1 (1994), 119-123.  doi: 10.1007/BF01193947.  Google Scholar [12] E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar [13] A. Farina, L. Montoro, G. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1-22.  doi: 10.1016/j.anihpc.2013.09.005.  Google Scholar [14] A. Farina, L. Montoro and B. Sciunzi, Monotonicity and one-dimensional symmetry for solutions of $-\Delta_pu = f(u)$ in half-spaces, Calc. Var. Partial Differ. Equ., 43 (2012), 123-145.  doi: 10.1007/s00526-011-0405-z.  Google Scholar [15] A. Farina, L. Montoro and B. Sciunzi, Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces, Math. Ann., 357 (2013), 855-893.  doi: 10.1007/s00208-013-0919-0.  Google Scholar [16] A. Farina, L. Montoro and B. Sciunzi, Monotonicity in half-space of positive solutions to $-\Delta_pu = f(u)$ in the case $p>2$, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (2017), 1207-1229.   Google Scholar [17] A. Farina and E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal., 195 (2010), 1025-1058.  doi: 10.1007/s00205-009-0227-8.  Google Scholar [18] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar [19] T. Kilpeläinen, H. Shahgholian and X. Zhong, Growth estimates through scaling for quasilinear partial differential equations, Ann. Acad. Sci. Fenn. Math., 32 (2007), 595-599.   Google Scholar [20] L. Montoro, G. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differ. Equ., 20 (2015), 717-740.   Google Scholar [21] P. Pucci and J. Serrin, The maximum principle, vol. 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007.  Google Scholar [22] B. Sciunzi, Classification of positive $\mathcal{D}^{1, p}(\mathbb{R}^N)$-solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, Adv. Math., 291 (2016), 12-23.  doi: 10.1016/j.aim.2015.12.028.  Google Scholar [23] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar [24] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.  Google Scholar [25] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differ. Equ., 42 (1981), 400-413.  doi: 10.1016/0022-0396(81)90113-3.  Google Scholar [26] J. Vétois, A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations, J. Differ. Equ., 260 (2016), 149-161.  doi: 10.1016/j.jde.2015.08.041.  Google Scholar
 [1] Giuseppe Riey, Berardino Sciunzi. One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1157-1166. doi: 10.3934/cpaa.2012.11.1157 [2] Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022 [3] Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045 [4] Dumitru Motreanu. Three solutions with precise sign properties for systems of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 831-843. doi: 10.3934/dcdss.2012.5.831 [5] Lynnyngs Kelly Arruda, Francisco Odair de Paiva, Ilma Marques. A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems. Conference Publications, 2011, 2011 (Special) : 112-116. doi: 10.3934/proc.2011.2011.112 [6] João Marcos do Ó, Sebastián Lorca, Justino Sánchez, Pedro Ubilla. Positive radial solutions for some quasilinear elliptic systems in exterior domains. Communications on Pure & Applied Analysis, 2006, 5 (3) : 571-581. doi: 10.3934/cpaa.2006.5.571 [7] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 [8] Meng Qu, Ping Li, Liu Yang. Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065 [9] Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307 [10] Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745 [11] Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051 [12] Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079 [13] Zhenjie Li, Chunqin Zhou. Radial symmetry of nonnegative solutions for nonlinear integral systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021201 [14] Toshiko Ogiwara, Hiroshi Matano. Monotonicity and convergence results in order-preserving systems in the presence of symmetry. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 1-34. doi: 10.3934/dcds.1999.5.1 [15] Marcos L. M. Carvalho, Edcarlos D. Silva, Claudiney Goulart, Carlos A. Santos. Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4401-4432. doi: 10.3934/cpaa.2020201 [16] Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 [17] Valentin Butuzov, Nikolay Nefedov, Oleh Omel'chenko, Lutz Recke. Boundary layer solutions to singularly perturbed quasilinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021226 [18] Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083 [19] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [20] Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151

2020 Impact Factor: 1.916

Article outline