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Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs
The regularity lifting methods for nonnegative solutions of Lane-Emden system
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China |
$ \begin{equation*} \begin{cases} -\Delta u = v^p\\ -\Delta v = u^q \end{cases} \mbox{ in } \mathbb{R}^n. \end{equation*} $ |
$ (\bar{u},\bar{v}) $ |
$ (\bar{u},\bar{v}) $ |
$ (\bar{u},\bar{v})\in L_{loc}^{r_0}(\mathbb{R}^n)\times L_{loc}^{s_0}(\mathbb{R}^n) $ |
$ r_0 $ |
$ s_0 $ |
$ (\bar{u},\bar{v})\in L_{loc}^r(\mathbb{R}^n)\times L_{loc}^s(\mathbb{R}^n) $ |
$ r $ |
$ s $ |
$ L^\infty(\mathbb{R}^n)\times L^\infty(\mathbb{R}^n). $ |
References:
[1] |
W. Chen and C. Li,
An integral system and the LaneEmden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[2] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, American Institute of Mathematical Sciences, Springfield, MO, 2010. |
[3] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[4] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019. |
[5] |
Z. Cheng and G. Huang,
A Liouville theorem for the subcritical Lane-Emden system, Discrete Contin. Dyn. Syst., 39 (2019), 1359-1377.
doi: 10.3934/dcds.2019058. |
[6] |
L. Evans, Partial Differential Equations, Wadsworth and Brooks/cole Mathematics, 2010. |
[7] |
D. G. de Figueiredo and P. L. Felmer,
A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.
|
[8] |
C. Li, Z. Wu and H. Xu,
Maximum principles and Bocher type theorems, Proc. Natl. Acad. Sci., 115 (2018), 6976-6979.
doi: 10.1073/pnas.1804225115. |
[9] |
C. Ma, W. Chen and C. Li,
Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[10] |
L. Ma and D. Chen,
A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
doi: 10.3934/cpaa.2006.5.855. |
[11] |
E. Mitidieri,
Nonexistence of positive solutions of semilinear elliptic systems in ${\textbf{R}}^N$, Differ. Integral Equ., 9 (1996), 465-479.
|
[12] |
E. Mitidieri,
A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri and S. I. Pokhozhaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.
|
[14] |
P. Poláčik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[15] |
W. Reichel and H. Zou,
Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[16] |
J. Serrin and H. Zou,
Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380.
|
[17] |
E. M. Stein and G. Weiss,
Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
[18] |
J. Serrin and H. Zou,
Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[19] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.
|
[20] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[21] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differ. Integral Equ., 8 (1995), 1245-1258.
|
show all references
References:
[1] |
W. Chen and C. Li,
An integral system and the LaneEmden conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[2] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, American Institute of Mathematical Sciences, Springfield, MO, 2010. |
[3] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[4] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019. |
[5] |
Z. Cheng and G. Huang,
A Liouville theorem for the subcritical Lane-Emden system, Discrete Contin. Dyn. Syst., 39 (2019), 1359-1377.
doi: 10.3934/dcds.2019058. |
[6] |
L. Evans, Partial Differential Equations, Wadsworth and Brooks/cole Mathematics, 2010. |
[7] |
D. G. de Figueiredo and P. L. Felmer,
A Liouville-type theorem for elliptic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.
|
[8] |
C. Li, Z. Wu and H. Xu,
Maximum principles and Bocher type theorems, Proc. Natl. Acad. Sci., 115 (2018), 6976-6979.
doi: 10.1073/pnas.1804225115. |
[9] |
C. Ma, W. Chen and C. Li,
Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699.
doi: 10.1016/j.aim.2010.07.020. |
[10] |
L. Ma and D. Chen,
A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859.
doi: 10.3934/cpaa.2006.5.855. |
[11] |
E. Mitidieri,
Nonexistence of positive solutions of semilinear elliptic systems in ${\textbf{R}}^N$, Differ. Integral Equ., 9 (1996), 465-479.
|
[12] |
E. Mitidieri,
A Rellich type identity and applications, Commun. Partial Differ. Equ., 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[13] |
E. Mitidieri and S. I. Pokhozhaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.
|
[14] |
P. Poláčik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[15] |
W. Reichel and H. Zou,
Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[16] |
J. Serrin and H. Zou,
Existence of positive solutions of the Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 369-380.
|
[17] |
E. M. Stein and G. Weiss,
Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.
doi: 10.1512/iumj.1958.7.57030. |
[18] |
J. Serrin and H. Zou,
Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[19] |
J. Serrin and H. Zou,
Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.
|
[20] |
P. Souplet,
The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.
doi: 10.1016/j.aim.2009.02.014. |
[21] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differ. Integral Equ., 8 (1995), 1245-1258.
|

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