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The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function
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 |
$ \begin{equation*} -Qu = e^u ~~~~\mbox{in}~~~~ \Omega\subset \mathbb{R}^{N}, \end{equation*} $ |
$ Q $ |
$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $ |
$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}} $ |
$ F: \mathbb{R}^{N}\rightarrow[0, +\infty) $ |
$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $ |
$ \Omega $ |
$ N-10 $ |
$ N<10 $ |
$ 2<N<10 $ |
$ N = 2 $ |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
A. Cianchi and P. Salani,
Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.
doi: 10.1007/s00208-009-0386-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
M. Fazly,
Entire solutions of quasilinear symmetric systems, Indiana Univ. Math. J., 66 (2017), 361-400.
doi: 10.1512/iumj.2017.66.6017. |
[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. |
[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. |
[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. |
[24] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998. |
[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. |
[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. |
[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. |
[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. |
[29] |
F. Pacard,
Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math, 79 (1993), 161-172.
doi: 10.1007/BF02568335. |
[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. |
[31] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math, 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[6] |
A. Cianchi and P. Salani,
Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.
doi: 10.1007/s00208-009-0386-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
M. Fazly,
Entire solutions of quasilinear symmetric systems, Indiana Univ. Math. J., 66 (2017), 361-400.
doi: 10.1512/iumj.2017.66.6017. |
[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. |
[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. |
[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. |
[24] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998. |
[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. |
[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. |
[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. |
[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. |
[29] |
F. Pacard,
Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math, 79 (1993), 161-172.
doi: 10.1007/BF02568335. |
[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. |
[31] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math, 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[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. |
[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. |
[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. |
[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. |
[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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