March  2022, 21(3): 1027-1048. doi: 10.3934/cpaa.2022008

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 Published  March 2022 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 and Applied Analysis, 2022, 21 (3) : 1027-1048. 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.

[2]

H. BerestyckiL. 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.

[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.

[4]

H. BerestyckiL. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94. 

[5]

M. F. Bidaut-VéronR. 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.

[6]

Z. ChenC. 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[13]

A. FarinaL. MontoroG. 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.

[14]

A. FarinaL. 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.

[15]

A. FarinaL. 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.

[16]

A. FarinaL. 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. 

[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.

[18]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[19]

T. KilpeläinenH. Shahgholian and X. Zhong, Growth estimates through scaling for quasilinear partial differential equations, Ann. Acad. Sci. Fenn. Math., 32 (2007), 595-599. 

[20]

L. MontoroG. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differ. Equ., 20 (2015), 717-740. 

[21]

P. Pucci and J. Serrin, The maximum principle, vol. 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007.

[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.

[23]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.

[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.

[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.

[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.

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.

[2]

H. BerestyckiL. 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.

[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.

[4]

H. BerestyckiL. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 69-94. 

[5]

M. F. Bidaut-VéronR. 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.

[6]

Z. ChenC. 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[13]

A. FarinaL. MontoroG. 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.

[14]

A. FarinaL. 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.

[15]

A. FarinaL. 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.

[16]

A. FarinaL. 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. 

[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.

[18]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[19]

T. KilpeläinenH. Shahgholian and X. Zhong, Growth estimates through scaling for quasilinear partial differential equations, Ann. Acad. Sci. Fenn. Math., 32 (2007), 595-599. 

[20]

L. MontoroG. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differ. Equ., 20 (2015), 717-740. 

[21]

P. Pucci and J. Serrin, The maximum principle, vol. 73 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Basel, 2007.

[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.

[23]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.

[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.

[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.

[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.

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