doi: 10.3934/cpaa.2022008
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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. 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.  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. 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.   Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   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. 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.  Google Scholar

[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.   Google Scholar

[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.   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. 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.  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. 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.   Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   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. 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.  Google Scholar

[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.   Google Scholar

[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.   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

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