American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data
  • DCDS Home
  • This Issue
  • Next Article
    Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching
September  2021, 41(9): 4185-4206. doi: 10.3934/dcds.2021033

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

Mostafa Fazly 1,, and  Yuan Li 2,

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  September 2021 Early access  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.
Keywords: Quasilinear equation, partial regularity, Liouville theorem, stable solutions, Hausdorff dimension.
Mathematics Subject Classification: 35J62, 35B65, 35B08.
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. 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]

Juan Dávila, Olivier Goubet. Partial regularity for a Liouville system. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2495-2503. doi: 10.3934/dcds.2014.34.2495
[2]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865
[3]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016
[4]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
[5]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
[6]

Zongming Guo, Juncheng Wei. Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2561-2580. doi: 10.3934/dcds.2014.34.2561
[7]

Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023
[8]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4397-4419. doi: 10.3934/dcds.2021041
[9]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
[10]

Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549
[11]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565
[12]

Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025
[13]

Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011
[14]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
[15]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248
[16]

Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Y. S. Hamed. Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021083
[17]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503
[18]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[19]

Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403
[20]

Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5189-5202. doi: 10.3934/dcds.2013.33.5189

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (142)
  • HTML views (199)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences