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Partial regularity for weak solutions of anisotropic Lane-Emden equation. (English) Zbl 1479.35422

Summary: We study positive weak solutions of the quasilinear Lane-Emden equation \[ -Qu=u^{\alpha}\quad\text{in}\quad\Omega\subset\mathbb{R}^n, \] where \(\alpha\geq\frac{n+2}{n-2}\), for \(n\geq 3\), is supercritical 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, \(F_{\xi_i}=\frac{\partial F}{\partial \xi_i}\) and \(F: \mathbb{R}^n\rightarrow [0,+\infty)\) is a convex function of \(C^2(\mathbb{R}^n\setminus\{0\})\), that satisfies positive homogeneity of first order and other certain assumptions. We prove that the Hausdorff dimension of singular set of \(u\) is less than \(n-2\frac{\alpha+1}{\alpha-1}\).

MSC:

35J62 Quasilinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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[1] Almgren, Fred; Taylor, Jean E., Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., 42, 1, 1-22 (1995) · Zbl 0867.58020
[2] Almgren, Fred; Taylor, Jean E.; Wang, Lihe, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31, 2, 387-438 (1993) · Zbl 0783.35002 · doi:10.1137/0331020
[3] Alvino, Angelo; Ferone, Vincenzo; Trombetti, Guido; Lions, Pierre-Louis, Convex symmetrization and applications, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 14, 2, 275-293 (1997) · Zbl 0877.35040 · doi:10.1016/S0294-1449(97)80147-3
[4] Amar, Micol; Bellettini, Giovanni, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 11, 1, 91-133 (1994) · Zbl 0842.49016 · doi:10.1016/S0294-1449(16)30197-4
[5] Campanato, Sergio, Equazioni ellittiche del \({\text{II}}\deg\) ordine espazi \({\mathfrak{L}}^{(2,\lambda )} \), Ann. Mat. Pura Appl. (4), 69, 321-381 (1965) · Zbl 0145.36603 · doi:10.1007/BF02414377
[6] Cianchi, Andrea; Salani, Paolo, Overdetermined anisotropic elliptic problems, Math. Ann., 345, 4, 859-881 (2009) · Zbl 1179.35107 · doi:10.1007/s00208-009-0386-9
[7] Cozzi, Matteo; Farina, Alberto; Valdinoci, Enrico, Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Adv. Math., 293, 343-381 (2016) · Zbl 1357.35153 · doi:10.1016/j.aim.2016.02.014
[8] Cozzi, Matteo; Farina, Alberto; Valdinoci, Enrico, Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations, Comm. Math. Phys., 331, 1, 189-214 (2014) · Zbl 1303.49001 · doi:10.1007/s00220-014-2107-9
[9] D\'{a}vila, Juan; Dupaigne, Louis; Wang, Kelei; Wei, Juncheng, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, Adv. Math., 258, 240-285 (2014) · Zbl 1317.35054 · doi:10.1016/j.aim.2014.02.034
[10] Farina, Alberto; Valdinoci, Enrico, Gradient bounds for anisotropic partial differential equations, Calc. Var. Partial Differential Equations, 49, 3-4, 923-936 (2014) · Zbl 1288.35131 · doi:10.1007/s00526-013-0605-9
[11] Fazly, Mostafa; Li, Yuan, Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations, Discrete Contin. Dyn. Syst., 41, 9, 4185-4206 (2021) · Zbl 1466.35188 · doi:10.3934/dcds.2021033
[12] Ferone, Vincenzo; Kawohl, Bernd, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137, 1, 247-253 (2009) · Zbl 1161.35017 · doi:10.1090/S0002-9939-08-09554-3
[13] Fonseca, Irene; M\"{u}ller, Stefan, A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. Edinburgh Sect. A, 119, 1-2, 125-136 (1991) · Zbl 0752.49019 · doi:10.1017/S0308210500028365
[14] GT David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1998) · Zbl 0361.35003
[15] Hardt, Robert; Kinderlehrer, David; Lin, Fang-Hua, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys., 105, 4, 547-570 (1986) · Zbl 0611.35077
[16] Lin, Fanghua; Yang, Xiaoping, Geometric measure theory-an introduction, Advanced Mathematics (Beijing/Boston) 1, x+237 pp. (2002), Science Press Beijing, Beijing; International Press, Boston, MA · Zbl 1074.49011
[17] Pacard, Frank, A note on the regularity of weak solutions of \(-\Delta u=u^\alpha\) in \({\mathbf{R}}^n,\ n\geq 3\), Houston J. Math., 18, 4, 621-632 (1992) · Zbl 0819.35045
[18] Pacard, Frank, Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math., 79, 2, 161-172 (1993) · Zbl 0811.35011 · doi:10.1007/BF02568335
[19] Pacard, Frank, Convergence and partial regularity for weak solutions of some nonlinear elliptic equation: the supercritical case, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire, 11, 5, 537-551 (1994) · Zbl 0837.35026 · doi:10.1016/S0294-1449(16)30177-9
[20] Ren, Xiaofeng; Wei, Juncheng, Counting peaks of solutions to some quasilinear elliptic equations with large exponents, J. Differential Equations, 117, 1, 28-55 (1995) · Zbl 0814.35040 · doi:10.1006/jdeq.1995.1047
[21] Wang, Guofang; Xia, Chao, A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal., 199, 1, 99-115 (2011) · Zbl 1232.35103 · doi:10.1007/s00205-010-0323-9
[22] Wang, Guofang; Xia, Chao, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations, 252, 2, 1668-1700 (2012) · Zbl 1233.35053 · doi:10.1016/j.jde.2011.08.001
[23] Wang, Kelei, Partial regularity of stable solutions to the Emden equation, Calc. Var. Partial Differential Equations, 44, 3-4, 601-610 (2012) · Zbl 1245.35044 · doi:10.1007/s00526-011-0446-3
[24] W George Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Aufl\"osung der Kristallfl\"achen, Z. Krist 34 (1901), 449-530.
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