# American Institute of Mathematical Sciences

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 , respectively. We also provide an explicit solution that is stable outside a compact set in two dimensions $ N = 2 $. In addition, we present similar Liouville theorems for the related equations with power-type nonlinearities. Citation: Mostafa Fazly, Yuan Li. Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2021033 ##### References:  [1] A. Alvino, V. Ferone, G. Trombetti and P. L. Lions, Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 14 (1997), 275-293. doi: 10.1016/S0294-1449(97)80147-3. Google Scholar [2] M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 11 (1994), 91-133. doi: 10.1016/S0294-1449(16)30197-4. Google Scholar [3] W. W. Ao and W. Yang, On the classification of solutions of cosmic strings equation, Ann. Mat. Pura Appl., 198 (2019), 2183-2193. doi: 10.1007/s10231-019-00861-w. Google Scholar [4] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of$-\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. Caffarelli, N. 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. Cozzi, A. 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. Cozzi, A. 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. Davila, L. Dupaigne, K. 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. Esposito, N. 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. Alvino, V. Ferone, G. 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. Caffarelli, N. 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. Cozzi, A. 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. Cozzi, A. 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. Davila, L. Dupaigne, K. 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. Esposito, N. 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  [1] Wentao Huang, Jianlin Xiang. 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