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Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group
Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data
1. | School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, Gansu 730030, China |
2. | Key Laboratory of Streaming Data Computing Technologies and Application, Northwest Minzu University, Lanzhou, Gansu 730030, China |
3. | Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China |
$ u $ |
$ \begin{align*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{(p-1)\theta}}\right)+|u|^{q-1}u = \lambda, \; &x\in\Omega, \\ u = 0, \; &x\in\partial\Omega, \end{array} \right. \end{align*} $ |
$ \begin{align*} q>\frac{r(p-1)[1+\theta(p-1)]}{r-p}, \end{align*} $ |
$ \Omega $ |
$ \mathbb{R}^N(N>2) $ |
$ 1<p<N $ |
$ q>1 $ |
$ 0\leq\theta<1 $ |
$ \lambda $ |
$ r $ |
$ (p<r\leq N) $ |
References:
[1] |
A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura. Appl. (4), 182 (2003), 53-79.
doi: 10.1007/s10231-002-0056-y. |
[2] |
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An L1 theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273. |
[3] |
P. Bénilan, H. Brézis and M. Crandall, A semilinear equation in $L^1(\mathbb{R}^N)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523-555. |
[4] |
L. Boccardo,
Some elliptic problems with degenerate coercivity, Adv. Nonlinear Stud., 6 (2006), 1-12.
doi: 10.1515/ans-2006-0101. |
[5] |
L. Boccardo,
Some cases of weak continuity in nonlinear Dirichlet problems, J. Funct. Anal., 277 (2019), 3673-3687.
doi: 10.1016/j.jfa.2019.05.020. |
[6] |
L. Boccardo and H. Brézis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 6 (2003), 521-530. |
[7] |
L. Boccardo, G. Croce and L. Orsina,
Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439.
doi: 10.1007/s00229-011-0473-6. |
[8] |
L. Boccardo, T. Gallouët and L. Orsina,
Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551.
doi: 10.1016/S0294-1449(16)30113-5. |
[9] |
H. Brézis, Nonlinear elliptic equations involving measures, in Contributions to Nonlinear Partial Differential Equations, Res. Notes Math., 89, Pitman, Boston, MA, 1983, 82–89. |
[10] |
G. Cirmi,
On the existence of solutions to non-linear degenerate elliptic equations with measures data, Ricerche Mat., 42 (1993), 315-329.
|
[11] |
G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions for elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808. |
[12] |
D. Giachetti and M. Porzio,
Exitence results for some nonuniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257 (2001), 100-130.
doi: 10.1006/jmaa.2000.7324. |
[13] |
D. Giachetti and M. Porzio,
Elliptic equations with degenerate coercivity: Gradient regularity, Acta. Math. Sin. (Engl. Ser.), 19 (2003), 349-370.
doi: 10.1007/s10114-002-0235-1. |
[14] |
S. Huang,
Quasilinear elliptic equations with exponential nonlinearity and measure data, Math. Methods Appl. Sci., 43 (2020), 2883-2910.
doi: 10.1002/mma.6088. |
[15] |
S. Huang, T. Su, X. Du and X. Zhang,
Entropy solutions to noncoercive nonlinear elliptic equations with measure data, Electron. J. Differential Equations, 2019 (2019), 1-22.
|
[16] |
S. Huang and Q. Tian,
Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation, Comput. Math. Appl., 78 (2019), 1732-1738.
doi: 10.1016/j.camwa.2019.04.032. |
[17] |
S. Huang and Q. Tian, Harnack-type inequality for fractional elliptic equations with critical exponent, Math. Methods Appl. Sci., (2020), 1–18.
doi: 10.1002/mma.6280. |
[18] |
S. Huang, Q. Tian, J. Wang and J. Mu,
Stability for noncoercive elliptic equations, Electron. J. Differential Equations, 2016 (2016), 1-11.
|
[19] |
H.-F. Huo, Q. Yang and H. Xiang,
Dynamics of an edge-based SEIR model for sexually transmitted diseases, Math. Biosci. Eng., 17 (2020), 669-699.
doi: 10.3934/mbe.2020035. |
[20] |
J. Leray and J. L. Lions,
Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.
doi: 10.24033/bsmf.1617. |
[21] |
X. Li and S. Huang, Stability and bifurcation for a single-species model with delay weak kernel and constant rate harvesting, Complexity, 2019 (2019).
doi: 10.1155/2019/1810385. |
[22] |
L. Orsina and A. Prignet,
Non-existence of solutions for some nonlinear elliptic equations involving measures, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 167-187.
doi: 10.1017/S0308210500000093. |
[23] |
L. Orsina and A. Porretta,
Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data, Commum. Contemp. Math., 3 (2001), 259-285.
doi: 10.1142/S0219199701000378. |
[24] |
L. Orsina and A. Prignet,
Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data, J. Funct. Anal., 189 (2002), 549-566.
doi: 10.1006/jfan.2001.3846. |
[25] |
M. M. Porzio and F. Smarrazzo, Radon measure-valued solutions for some quasilinear degenerate elliptic equations, Ann. Mat. Pura. Appl. (4), 194 (2015), 495-532.
doi: 10.1007/s10231-013-0386-y. |
[26] |
Q. Tian and Y. Xu, Effect of the domain geometry on the solutions to fractional Brezis-Nirenberg problem, J. Funct. Spaces, 2019 (2019), 4pp.
doi: 10.1155/2019/1093804. |
[27] |
Y. Ye, H. Liu, Y. Wei, M. Ma and K. Zhang, Dynamic study of a predator-prey model with weak Allee effect and delay, Adv. Math. Phys., 2019 (2019), 15pp.
doi: 10.1155/2019/7296461. |
show all references
References:
[1] |
A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura. Appl. (4), 182 (2003), 53-79.
doi: 10.1007/s10231-002-0056-y. |
[2] |
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An L1 theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273. |
[3] |
P. Bénilan, H. Brézis and M. Crandall, A semilinear equation in $L^1(\mathbb{R}^N)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523-555. |
[4] |
L. Boccardo,
Some elliptic problems with degenerate coercivity, Adv. Nonlinear Stud., 6 (2006), 1-12.
doi: 10.1515/ans-2006-0101. |
[5] |
L. Boccardo,
Some cases of weak continuity in nonlinear Dirichlet problems, J. Funct. Anal., 277 (2019), 3673-3687.
doi: 10.1016/j.jfa.2019.05.020. |
[6] |
L. Boccardo and H. Brézis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 6 (2003), 521-530. |
[7] |
L. Boccardo, G. Croce and L. Orsina,
Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439.
doi: 10.1007/s00229-011-0473-6. |
[8] |
L. Boccardo, T. Gallouët and L. Orsina,
Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551.
doi: 10.1016/S0294-1449(16)30113-5. |
[9] |
H. Brézis, Nonlinear elliptic equations involving measures, in Contributions to Nonlinear Partial Differential Equations, Res. Notes Math., 89, Pitman, Boston, MA, 1983, 82–89. |
[10] |
G. Cirmi,
On the existence of solutions to non-linear degenerate elliptic equations with measures data, Ricerche Mat., 42 (1993), 315-329.
|
[11] |
G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions for elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808. |
[12] |
D. Giachetti and M. Porzio,
Exitence results for some nonuniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257 (2001), 100-130.
doi: 10.1006/jmaa.2000.7324. |
[13] |
D. Giachetti and M. Porzio,
Elliptic equations with degenerate coercivity: Gradient regularity, Acta. Math. Sin. (Engl. Ser.), 19 (2003), 349-370.
doi: 10.1007/s10114-002-0235-1. |
[14] |
S. Huang,
Quasilinear elliptic equations with exponential nonlinearity and measure data, Math. Methods Appl. Sci., 43 (2020), 2883-2910.
doi: 10.1002/mma.6088. |
[15] |
S. Huang, T. Su, X. Du and X. Zhang,
Entropy solutions to noncoercive nonlinear elliptic equations with measure data, Electron. J. Differential Equations, 2019 (2019), 1-22.
|
[16] |
S. Huang and Q. Tian,
Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation, Comput. Math. Appl., 78 (2019), 1732-1738.
doi: 10.1016/j.camwa.2019.04.032. |
[17] |
S. Huang and Q. Tian, Harnack-type inequality for fractional elliptic equations with critical exponent, Math. Methods Appl. Sci., (2020), 1–18.
doi: 10.1002/mma.6280. |
[18] |
S. Huang, Q. Tian, J. Wang and J. Mu,
Stability for noncoercive elliptic equations, Electron. J. Differential Equations, 2016 (2016), 1-11.
|
[19] |
H.-F. Huo, Q. Yang and H. Xiang,
Dynamics of an edge-based SEIR model for sexually transmitted diseases, Math. Biosci. Eng., 17 (2020), 669-699.
doi: 10.3934/mbe.2020035. |
[20] |
J. Leray and J. L. Lions,
Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.
doi: 10.24033/bsmf.1617. |
[21] |
X. Li and S. Huang, Stability and bifurcation for a single-species model with delay weak kernel and constant rate harvesting, Complexity, 2019 (2019).
doi: 10.1155/2019/1810385. |
[22] |
L. Orsina and A. Prignet,
Non-existence of solutions for some nonlinear elliptic equations involving measures, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 167-187.
doi: 10.1017/S0308210500000093. |
[23] |
L. Orsina and A. Porretta,
Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data, Commum. Contemp. Math., 3 (2001), 259-285.
doi: 10.1142/S0219199701000378. |
[24] |
L. Orsina and A. Prignet,
Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data, J. Funct. Anal., 189 (2002), 549-566.
doi: 10.1006/jfan.2001.3846. |
[25] |
M. M. Porzio and F. Smarrazzo, Radon measure-valued solutions for some quasilinear degenerate elliptic equations, Ann. Mat. Pura. Appl. (4), 194 (2015), 495-532.
doi: 10.1007/s10231-013-0386-y. |
[26] |
Q. Tian and Y. Xu, Effect of the domain geometry on the solutions to fractional Brezis-Nirenberg problem, J. Funct. Spaces, 2019 (2019), 4pp.
doi: 10.1155/2019/1093804. |
[27] |
Y. Ye, H. Liu, Y. Wei, M. Ma and K. Zhang, Dynamic study of a predator-prey model with weak Allee effect and delay, Adv. Math. Phys., 2019 (2019), 15pp.
doi: 10.1155/2019/7296461. |
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