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A multiparameter fractional Laplace problem with semipositone nonlinearity
Extremal solution and Liouville theorem for anisotropic elliptic equations
School of Mathematics, Hunan University, Changsha 410082, China |
$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $ |
$ \lambda>0 $ |
$ \Omega\subset\mathbb{R}^{N} $ |
$ N\geq2 $ |
$ 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}}(\xi) $ |
$ F: \mathbb{R}^{N}\rightarrow [0, +\infty) $ |
$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $ |
$ N\leq9 $ |
$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, $ |
$ \alpha>-2 $ |
$ N\geq2 $ |
$ F^{0} $ |
$ K: = \{x\in\mathbb{R}^{N}:F(x)<1\} $ |
$ 2\leq N<10+4\alpha $ |
$ 3\leq N<10+4\alpha^{-} $ |
$ \alpha^{-} = \min\{\alpha, 0\} $ |
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Brezis and L. Nirenberg,
Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.
doi: 10.1002/cpa.3160440808. |
[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. |
[9] |
A. Cianchi and P. Salani,
Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.
doi: 10.1007/s00208-009-0386-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. Le,
Low dimensional instability for quasilinear problems of weighted exponential nonlinearity, Math. Nachr., 291 (2018), 2288-2297.
doi: 10.1002/mana.201700260. |
[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. |
[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. |
[26] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[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. |
[28] |
G. Stampacchia,
Équations elliptiques du second ordre à coefficients discontinus, Séminaire Jean Leray, 3 (1963-1964), 1-77.
|
[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. |
[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. |
[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. |
[32] |
G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflung der Kristallflhen, Z. Krist, 34 (1901), 44930. |
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Brezis and L. Nirenberg,
Remarks on finding critical points, Comm. Pure Appl. Math., 44 (1991), 939-963.
doi: 10.1002/cpa.3160440808. |
[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. |
[9] |
A. Cianchi and P. Salani,
Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.
doi: 10.1007/s00208-009-0386-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
P. Le,
Low dimensional instability for quasilinear problems of weighted exponential nonlinearity, Math. Nachr., 291 (2018), 2288-2297.
doi: 10.1002/mana.201700260. |
[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. |
[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. |
[26] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[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. |
[28] |
G. Stampacchia,
Équations elliptiques du second ordre à coefficients discontinus, Séminaire Jean Leray, 3 (1963-1964), 1-77.
|
[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. |
[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. |
[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. |
[32] |
G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflung der Kristallflhen, Z. Krist, 34 (1901), 44930. |
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