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September  2021, 41(9): 4185-4206. doi: 10.3934/dcds.2021033

Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations

1. 

Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA

2. 

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China, Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA

* Corresponding author: Mostafa Fazly

Received  June 2020 Revised  January 2021 Published  February 2021

Fund Project: This work is part of the second author's Ph.D. dissertation and supported by a China Scholarship Council funding

We study the quasilinear elliptic equation
$ \begin{equation*} -Qu = e^u ~~~~\mbox{in}~~~~ \Omega\subset \mathbb{R}^{N}, \end{equation*} $
where the operator
$ Q $
, known as the Finsler-Laplacian (or anisotropic Laplacian) operator, is defined by
$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $
Here
$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}} $
and
$ F: \mathbb{R}^{N}\rightarrow[0, +\infty) $
is a convex function of
$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $
that satisfies certain assumptions. For a bounded domain
$ \Omega $
and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed
$ N-10 $
. For the case of entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions
$ N<10 $
and
$ 2<N<10 $
, respectively. We also provide an explicit solution that is stable outside a compact set in two dimensions
$ N = 2 $
. In addition, we present similar Liouville theorems for the related equations with power-type nonlinearities.
Citation: Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4185-4206. doi: 10.3934/dcds.2021033
References:
[1]

A. AlvinoV. FeroneG. Trombetti and P. L. Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 14 (1997), 275-293.  doi: 10.1016/S0294-1449(97)80147-3.  Google Scholar

[2]

M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 11 (1994), 91-133.  doi: 10.1016/S0294-1449(16)30197-4.  Google Scholar

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L. CaffarelliN. Garofalo and F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences, omm. Pure Appl. Math., 47 (1994), 1457-1473.  doi: 10.1002/cpa.3160471103.  Google Scholar

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A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.  doi: 10.1007/s00208-009-0386-9.  Google Scholar

[7]

G. Ciraolo, A. Figalli and A. Roncoroni, Symmetry results for critical anisotropic p-Laplacian equations in convex cones, Geom. Funct. Anal., 30 (2020), 770–803, arXiv: 1906.00622v1. doi: 10.1007/s00039-020-00535-3.  Google Scholar

[8]

M. CozziA. Farina and E. Valdinoci, Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Adv. Math., 293 (2016), 343-381.  doi: 10.1016/j.aim.2016.02.014.  Google Scholar

[9]

M. CozziA. Farina and E. Valdinoci, Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations, Comm. Math. Phys., 331 (2014), 189-214.  doi: 10.1007/s00220-014-2107-9.  Google Scholar

[10]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741.  Google Scholar

[11]

F. Dalio, Partial regularity for stationary solutions to Liouville-type equation in dimension $3$, Comm. Partial Differential Equations, 33 (2008), 1890-1910.  doi: 10.1080/03605300802402625.  Google Scholar

[12]

E. N. Dancer and A. Farina, On the classification of solutions of $-\Delta u = e^{u}$ on $\mathbb{R}^{N}$: stability outside a compact set and applications, Proc. Amer. Math. Soc, 137 (2009), 1333-1338.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

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E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 173-179.  doi: 10.1016/j.anihpc.2006.12.001.  Google Scholar

[14]

J. DavilaL. DupaigneK. L. Wang and J. C. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[15]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. doi: 10.1090/cln/020.  Google Scholar

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P. EspositoN. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math, 60 (2007), 1731-1768.  doi: 10.1002/cpa.20189.  Google Scholar

[17]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^{N}$, J. Math. Pures Appl, 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[18]

A. Farina, Stable solutions of $-\Delta u = e^{u}$ on $\mathbb{R}^{N}$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021.  Google Scholar

[19]

A. Farina and E. Valdinoci, Gradient bounds for anisotropic partial differential equations, Calc. Var. Partial Differential Equations, 49 (2014), 923-936.  doi: 10.1007/s00526-013-0605-9.  Google Scholar

[20]

M. Fazly, Entire solutions of quasilinear symmetric systems, Indiana Univ. Math. J., 66 (2017), 361-400.  doi: 10.1512/iumj.2017.66.6017.  Google Scholar

[21]

M. Fazly and H. Shahgholian, Monotonicity formulas for coupled elliptic gradient systems with applications, Adv. Nonlinear Anal, 9 (2020), 479-495.  doi: 10.1515/anona-2020-0010.  Google Scholar

[22]

V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc, 137 (2009), 247-253.  doi: 10.1090/S0002-9939-08-09554-3.  Google Scholar

[23]

I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 125-136.  doi: 10.1017/S0308210500028365.  Google Scholar

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998.  Google Scholar

[25]

Z. M. Guo and J. C. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math, 120 (2006), 193-209.  doi: 10.1007/s00229-006-0001-2.  Google Scholar

[26]

F. H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction. Advanced Mathematics (Beijing/Boston), vol. 1. Science Press/International Press, Boston/Beijing, 2002.  Google Scholar

[27]

A. Mercaldo, M. Sano and F. Takahashi, Finsler Hardy inequalities, Math. Nachr., 293 (2020), 2370–2398, arXiv: 1806.04901v2. doi: 10.1002/mana.201900117.  Google Scholar

[28]

L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, in: Partial Differential Equations and the Calculus of Variations, vol. II, in: Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhkäser Boston, Boston, MA, 1989,843–850.  Google Scholar

[29]

F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math, 79 (1993), 161-172.  doi: 10.1007/BF02568335.  Google Scholar

[30]

X. F. Ren and J. C. Wei, Counting peaks of solutions to some quasilinear elliptic equations with large exponents, J. Differ. Equations, 117 (1995), 28-55.  doi: 10.1006/jdeq.1995.1047.  Google Scholar

[31]

J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math, 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar

[32]

G. F. Wang and C. Xia, A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Rational Mech. Anal, 199 (2011), 99-115.  doi: 10.1007/s00205-010-0323-9.  Google Scholar

[33]

G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differ. Equations, 252 (2012), 1668-1700.  doi: 10.1016/j.jde.2011.08.001.  Google Scholar

[34]

K. L. Wang, Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 44 (2012), 601-610.  doi: 10.1007/s00526-011-0446-3.  Google Scholar

[35]

K. L. Wang, Erratum to: Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 47 (2013), 433-435.  doi: 10.1007/s00526-012-0565-5.  Google Scholar

[36]

G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflsung der Kristallflen, Z. Krist, 34 (1901), 449530. Google Scholar

show all references

References:
[1]

A. AlvinoV. FeroneG. Trombetti and P. L. Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 14 (1997), 275-293.  doi: 10.1016/S0294-1449(97)80147-3.  Google Scholar

[2]

M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 11 (1994), 91-133.  doi: 10.1016/S0294-1449(16)30197-4.  Google Scholar

[3]

W. W. Ao and W. Yang, On the classification of solutions of cosmic strings equation, Ann. Mat. Pura Appl., 198 (2019), 2183-2193.  doi: 10.1007/s10231-019-00861-w.  Google Scholar

[4]

H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u = V(x)e^{u}$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.  Google Scholar

[5]

L. CaffarelliN. Garofalo and F. Segala, A gradient bound for entire solutions of quasi-linear equations and its consequences, omm. Pure Appl. Math., 47 (1994), 1457-1473.  doi: 10.1002/cpa.3160471103.  Google Scholar

[6]

A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.  doi: 10.1007/s00208-009-0386-9.  Google Scholar

[7]

G. Ciraolo, A. Figalli and A. Roncoroni, Symmetry results for critical anisotropic p-Laplacian equations in convex cones, Geom. Funct. Anal., 30 (2020), 770–803, arXiv: 1906.00622v1. doi: 10.1007/s00039-020-00535-3.  Google Scholar

[8]

M. CozziA. Farina and E. Valdinoci, Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Adv. Math., 293 (2016), 343-381.  doi: 10.1016/j.aim.2016.02.014.  Google Scholar

[9]

M. CozziA. Farina and E. Valdinoci, Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations, Comm. Math. Phys., 331 (2014), 189-214.  doi: 10.1007/s00220-014-2107-9.  Google Scholar

[10]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741.  Google Scholar

[11]

F. Dalio, Partial regularity for stationary solutions to Liouville-type equation in dimension $3$, Comm. Partial Differential Equations, 33 (2008), 1890-1910.  doi: 10.1080/03605300802402625.  Google Scholar

[12]

E. N. Dancer and A. Farina, On the classification of solutions of $-\Delta u = e^{u}$ on $\mathbb{R}^{N}$: stability outside a compact set and applications, Proc. Amer. Math. Soc, 137 (2009), 1333-1338.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

[13]

E. N. Dancer, Finite Morse index solutions of exponential problems, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008), 173-179.  doi: 10.1016/j.anihpc.2006.12.001.  Google Scholar

[14]

J. DavilaL. DupaigneK. L. Wang and J. C. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258 (2014), 240-285.  doi: 10.1016/j.aim.2014.02.034.  Google Scholar

[15]

P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lecture Notes in Mathematics, 20. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. doi: 10.1090/cln/020.  Google Scholar

[16]

P. EspositoN. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math, 60 (2007), 1731-1768.  doi: 10.1002/cpa.20189.  Google Scholar

[17]

A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^{N}$, J. Math. Pures Appl, 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.  Google Scholar

[18]

A. Farina, Stable solutions of $-\Delta u = e^{u}$ on $\mathbb{R}^{N}$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021.  Google Scholar

[19]

A. Farina and E. Valdinoci, Gradient bounds for anisotropic partial differential equations, Calc. Var. Partial Differential Equations, 49 (2014), 923-936.  doi: 10.1007/s00526-013-0605-9.  Google Scholar

[20]

M. Fazly, Entire solutions of quasilinear symmetric systems, Indiana Univ. Math. J., 66 (2017), 361-400.  doi: 10.1512/iumj.2017.66.6017.  Google Scholar

[21]

M. Fazly and H. Shahgholian, Monotonicity formulas for coupled elliptic gradient systems with applications, Adv. Nonlinear Anal, 9 (2020), 479-495.  doi: 10.1515/anona-2020-0010.  Google Scholar

[22]

V. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc, 137 (2009), 247-253.  doi: 10.1090/S0002-9939-08-09554-3.  Google Scholar

[23]

I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 125-136.  doi: 10.1017/S0308210500028365.  Google Scholar

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998.  Google Scholar

[25]

Z. M. Guo and J. C. Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math, 120 (2006), 193-209.  doi: 10.1007/s00229-006-0001-2.  Google Scholar

[26]

F. H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction. Advanced Mathematics (Beijing/Boston), vol. 1. Science Press/International Press, Boston/Beijing, 2002.  Google Scholar

[27]

A. Mercaldo, M. Sano and F. Takahashi, Finsler Hardy inequalities, Math. Nachr., 293 (2020), 2370–2398, arXiv: 1806.04901v2. doi: 10.1002/mana.201900117.  Google Scholar

[28]

L. Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, in: Partial Differential Equations and the Calculus of Variations, vol. II, in: Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhkäser Boston, Boston, MA, 1989,843–850.  Google Scholar

[29]

F. Pacard, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math, 79 (1993), 161-172.  doi: 10.1007/BF02568335.  Google Scholar

[30]

X. F. Ren and J. C. Wei, Counting peaks of solutions to some quasilinear elliptic equations with large exponents, J. Differ. Equations, 117 (1995), 28-55.  doi: 10.1006/jdeq.1995.1047.  Google Scholar

[31]

J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math, 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar

[32]

G. F. Wang and C. Xia, A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Rational Mech. Anal, 199 (2011), 99-115.  doi: 10.1007/s00205-010-0323-9.  Google Scholar

[33]

G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differ. Equations, 252 (2012), 1668-1700.  doi: 10.1016/j.jde.2011.08.001.  Google Scholar

[34]

K. L. Wang, Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 44 (2012), 601-610.  doi: 10.1007/s00526-011-0446-3.  Google Scholar

[35]

K. L. Wang, Erratum to: Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 47 (2013), 433-435.  doi: 10.1007/s00526-012-0565-5.  Google Scholar

[36]

G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflsung der Kristallflen, Z. Krist, 34 (1901), 449530. Google Scholar

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