# American Institute of Mathematical Sciences

December  2021, 20(12): 4063-4082. doi: 10.3934/cpaa.2021144

## Extremal solution and Liouville theorem for anisotropic elliptic equations

 School of Mathematics, Hunan University, Changsha 410082, China

Received  February 2021 Revised  July 2021 Published  December 2021 Early access  August 2021

We study the quasilinear Dirichlet boundary problem
 $$$\nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases}$$$
where
 $\lambda>0$
is a parameter,
 $\Omega\subset\mathbb{R}^{N}$
(
 $N\geq2$
) is a bounded domain, and the operator
 $Q$
, known as Finsler-Laplacian or anisotropic Laplacian, is defined by
 $Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)).$
Here,
 $F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}}(\xi)$
and
 $F: \mathbb{R}^{N}\rightarrow [0, +\infty)$
is a convex function of
 $C^{2}(\mathbb{R}^{N}\setminus\{0\})$
, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if
 $N\leq9$
.
We also concern the Hénon type anisotropic Liouville equation,
 $-Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N},$
where
 $\alpha>-2$
,
 $N\geq2$
and
 $F^{0}$
is the support function of
 $K: = \{x\in\mathbb{R}^{N}:F(x)<1\}$
. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for
 $2\leq N<10+4\alpha$
and
 $3\leq N<10+4\alpha^{-}$
respectively, where
 $\alpha^{-} = \min\{\alpha, 0\}$
.
Citation: Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4063-4082. doi: 10.3934/cpaa.2021144
##### References:
 [1] F. Almgren and J. E. Taylor, Flat flow is motion by cristalline curvature for curves with cnstalline energies, J. Differ. Geom., 42 (1995), 1-22.   Google Scholar [2] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-437.  doi: 10.1137/0331020.  Google Scholar [3] 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 [4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [5] 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 [6] M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783.  doi: 10.1007/s00033-003-3209-y.  Google Scholar [7] H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.  doi: 10.1002/cpa.3160440808.  Google Scholar [8] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar [9] A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.  doi: 10.1007/s00208-009-0386-9.  Google Scholar [10] 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 [11] 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 [12] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problem, Arch. Rational Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741.  Google Scholar [13] 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 [14] F. Della Pietra and N. Gavitone, Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators, Math. Nachr., 287 (2014), 194-209.  doi: 10.1002/mana.201200296.  Google Scholar [15] 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 [16] A. Farina and E. Valdinoci, Gradient bounds for anisotropic partial differential equations, Calc. Var. Partial Differ. Equ., 49 (2014), 923-936.  doi: 10.1007/s00526-013-0605-9.  Google Scholar [17] M. Fazly and Y. Li, Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations, Discrete Contin. Dyn. Syst., 41 (2021), 4185-4206.  doi: 10.3934/dcds.2021033.  Google Scholar [18] 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 [19] G. M. Figueiredo and J. R. Silva, Solutions to an anisotropic system via subsupersolution method and Mountain Pass Theorem, Electronic Journal Quality Theory in Differential Equations, 46 (2019), 1-13.  doi: 10.14232/ejqtde.2019.1.46.  Google Scholar [20] 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 [21] J. Garcia Azorero and I. Peral Alonso, On an Emden-Fowler type equation, Nonlinear Anal., 18 (1992), 1085-1097.  doi: 10.1016/0362-546X(92)90197-M.  Google Scholar [22] J. Garcia Azorero, I. Peral Alonso and J. P. Puel, Quasilinear problems with exponential growth in the reaction term, Nonlinear Anal., 22 (1994), 481-498.  doi: 10.1016/0362-546X(94)90169-4.  Google Scholar [23] P. Le, Low dimensional instability for quasilinear problems of weighted exponential nonlinearity, Math. Nachr., 291 (2018), 2288-2297.  doi: 10.1002/mana.201700260.  Google Scholar [24] F. Mignot and J. P. Puel, Sur une class de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differ. Equ., 5 (1980), 791-836.  doi: 10.1080/03605308008820155.  Google Scholar [25] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar [26] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar [27] J. Serrin, On the strong maximum principle for quasilinear second order differential inequalities, J. Funct. Anal., 5 (1970), 184-193.  doi: 10.1016/0022-1236(70)90024-8.  Google Scholar [28] G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus, Séminaire Jean Leray, 3 (1963-1964), 1-77.   Google Scholar [29] C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, . Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017.  Google Scholar [30] 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 [31] G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differ. Equ., 252 (2012), 1668-1700.  doi: 10.1016/j.jde.2011.08.001.  Google Scholar [32] G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflung der Kristallflhen, Z. Krist, 34 (1901), 44930. Google Scholar

show all references

##### References:
 [1] F. Almgren and J. E. Taylor, Flat flow is motion by cristalline curvature for curves with cnstalline energies, J. Differ. Geom., 42 (1995), 1-22.   Google Scholar [2] F. Almgren, J. E. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-437.  doi: 10.1137/0331020.  Google Scholar [3] 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 [4] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [5] 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 [6] M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54 (2003), 771-783.  doi: 10.1007/s00033-003-3209-y.  Google Scholar [7] H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.  doi: 10.1002/cpa.3160440808.  Google Scholar [8] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar [9] A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.  doi: 10.1007/s00208-009-0386-9.  Google Scholar [10] 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 [11] 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 [12] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problem, Arch. Rational Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741.  Google Scholar [13] 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 [14] F. Della Pietra and N. Gavitone, Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators, Math. Nachr., 287 (2014), 194-209.  doi: 10.1002/mana.201200296.  Google Scholar [15] 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 [16] A. Farina and E. Valdinoci, Gradient bounds for anisotropic partial differential equations, Calc. Var. Partial Differ. Equ., 49 (2014), 923-936.  doi: 10.1007/s00526-013-0605-9.  Google Scholar [17] M. Fazly and Y. Li, Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations, Discrete Contin. Dyn. Syst., 41 (2021), 4185-4206.  doi: 10.3934/dcds.2021033.  Google Scholar [18] 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 [19] G. M. Figueiredo and J. R. Silva, Solutions to an anisotropic system via subsupersolution method and Mountain Pass Theorem, Electronic Journal Quality Theory in Differential Equations, 46 (2019), 1-13.  doi: 10.14232/ejqtde.2019.1.46.  Google Scholar [20] 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 [21] J. Garcia Azorero and I. Peral Alonso, On an Emden-Fowler type equation, Nonlinear Anal., 18 (1992), 1085-1097.  doi: 10.1016/0362-546X(92)90197-M.  Google Scholar [22] J. Garcia Azorero, I. Peral Alonso and J. P. Puel, Quasilinear problems with exponential growth in the reaction term, Nonlinear Anal., 22 (1994), 481-498.  doi: 10.1016/0362-546X(94)90169-4.  Google Scholar [23] P. Le, Low dimensional instability for quasilinear problems of weighted exponential nonlinearity, Math. Nachr., 291 (2018), 2288-2297.  doi: 10.1002/mana.201700260.  Google Scholar [24] F. Mignot and J. P. Puel, Sur une class de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differ. Equ., 5 (1980), 791-836.  doi: 10.1080/03605308008820155.  Google Scholar [25] W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar [26] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar [27] J. Serrin, On the strong maximum principle for quasilinear second order differential inequalities, J. Funct. Anal., 5 (1970), 184-193.  doi: 10.1016/0022-1236(70)90024-8.  Google Scholar [28] G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus, Séminaire Jean Leray, 3 (1963-1964), 1-77.   Google Scholar [29] C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, . Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017.  Google Scholar [30] 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 [31] G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differ. Equ., 252 (2012), 1668-1700.  doi: 10.1016/j.jde.2011.08.001.  Google Scholar [32] G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflung der Kristallflhen, Z. Krist, 34 (1901), 44930. Google Scholar
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