Proof of the Hénon-Lane-Emden conjecture in \(\mathbb{R}^3\). (English) Zbl 1410.35040
This paper deals with the Hénon-Lane-Emden conjecture \(C\) for the semilinear elliptic system
\[
\begin{aligned} -\Delta u &=|x|^a v^p,\\ -\Delta v &=|x|^b u^q,\end{aligned}\quad x\in\mathbb{R}^N,\tag{1}
\]
\(N\geq 2\), \(p,q>0\), \(a,b\in\mathbb{R}\).
The pair \((p,q)\) is called subcritical if \(\frac{N+a}{p+1}+\frac{N+b}{q+1}>N-2\). \(C\) claims that the system (1) has no positive classical solutions \(u\), \(v\), i.e., \(u>0\), \(v>0\) in \(\mathbb{R}^N\setminus 0\) for subcritical \((p,q)\).
In Theorem 1.1. this conjecture is completely solved for \(N=3\). As a corollary the scalar equation \(-\Delta u=|x|u^p\) has no positive classical solutions for \(N=3\) when \(p\) lies below the Hardy-Sobolev exponent. This way the conjecture of Phan-Souplet is proved in \(\mathbb{R}^3\). We note that in the previous results no growth or decay conditions are imposed on the solutions.
The pair \((p,q)\) is called subcritical if \(\frac{N+a}{p+1}+\frac{N+b}{q+1}>N-2\). \(C\) claims that the system (1) has no positive classical solutions \(u\), \(v\), i.e., \(u>0\), \(v>0\) in \(\mathbb{R}^N\setminus 0\) for subcritical \((p,q)\).
In Theorem 1.1. this conjecture is completely solved for \(N=3\). As a corollary the scalar equation \(-\Delta u=|x|u^p\) has no positive classical solutions for \(N=3\) when \(p\) lies below the Hardy-Sobolev exponent. This way the conjecture of Phan-Souplet is proved in \(\mathbb{R}^3\). We note that in the previous results no growth or decay conditions are imposed on the solutions.
Reviewer: Petar Popivanov (Sofia)
MSC:
| 35J61 | Semilinear elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
References:
| [1] | Bidaut-Véron, M. F.; Giacomini, H., A new dynamical approach of Emden-Fowler equations and systems, Adv. Differential Equations, 15, 11-12, 1033-1082, (2010) · Zbl 1230.34021 |
| [2] | Birindelli, I.; Mitidieri, E., Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128, 1217-1247, (1998) · Zbl 0919.35023 |
| [3] | Bonheure, D.; Moreirados Santos, E.; Ramos, M., Symmetry and symmetry breaking for ground state solutions of some strongly coupled elliptic systems, J. Funct. Anal., 264, 1, 62-96, (2013) · Zbl 1278.35069 |
| [4] | Busca, J.; Manásevich, R., A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51, 37-51, (2002) · Zbl 1033.35032 |
| [5] | Byeon, J.; Wang, Z. Q., On the Hénon equation: asymptotic profile of ground states, II, J. Differential Equations, 216, 78-108, (2005) · Zbl 1114.35070 |
| [6] | Byeon, J.; Wang, Z. Q., On the Hénon equation: asymptotic profile of ground states, I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 803-828, (2006) · Zbl 1114.35071 |
| [7] | Calanchi, M.; Ruf, B., Radial and non radial solutions for Hardy-Hénon type elliptic systems, Calc. Var. Partial Differential Equations, 38, 1-2, 111-133, (2010) · Zbl 1193.35035 |
| [8] | Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63, 3, 615-622, (1991) · Zbl 0768.35025 |
| [9] | Chen, W.; Li, C., An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 24, 4, 1167-1184, (2009) · Zbl 1176.35067 |
| [10] | Cheng, Z.; Huang, G.; Li, C., A Liouville theorem for subcritical Lane-Emden system, (2014) |
| [11] | Clement, Ph.; de Figueiredo, D. G.; Mitidieri, E., Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17, 923-940, (1992) · Zbl 0818.35027 |
| [12] | Dupaigne, L.; Ponce, A. C., Singularities of positive supersolutions in elliptic pdes, Selecta Math. (N.S.), 10, 3, 341-358, (2004) · Zbl 1133.35335 |
| [13] | de Figueiredo, D. G., Semilinear elliptic systems, (Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, 1997, (1998), World Sci. Publishing River Edge, NJ), 122-152 · Zbl 0955.35020 |
| [14] | Fazly, M.; Ghoussoub, N., On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34, 6, 2513-2533, (2014) · Zbl 1285.35024 |
| [15] | de Figueiredo, D. G.; Felmer, P., A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 21, 387-397, (1994) · Zbl 0820.35042 |
| [16] | de Figueiredo, D. G.; Sirakov, B., Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems, Math. Ann., 333, 231-260, (2005) · Zbl 1165.35360 |
| [17] | Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34, 4, 525-598, (1981) · Zbl 0465.35003 |
| [18] | Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, Classics Math., (2001), Springer-Verlag Berlin, reprint of the 1998 edition · Zbl 1042.35002 |
| [19] | Lin, C. S., A classification of solutions of a conformally invariant fourth order equation in \(\mathbb{R}^n\), Comment. Math. Helv., 73, 206-231, (1998) · Zbl 0933.35057 |
| [20] | Mitidieri, E., A Rellich type identity and applications, Comm. Partial Differential Equations, 18, 125-151, (1993) · Zbl 0816.35027 |
| [21] | Mitidieri, E., Non-existence of positive solutions of semilinear elliptic systems in \(\mathbb{R}^N\), Differential Integral Equations, 9, 3, 465-479, (1996) · Zbl 0848.35034 |
| [22] | Phan, Q. H.; Souplet, P., Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differential Equations, 252, 3, 2544-2562, (2012) · Zbl 1233.35093 |
| [23] | Poláčik, P.; Quittner, P.; Souplet, P., Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. elliptic equations and systems, Duke Math. J., 139, 3, 555-579, (2007) · Zbl 1146.35038 |
| [24] | Quittner, P.; Souplet, P., Superlinear parabolic problems. blow-up, global existence and steady states, Birkhäuser Adv. Texts, (2007), Springer Berlin · Zbl 1128.35003 |
| [25] | Reichel, W.; Zou, H., Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161, 219-243, (2000) · Zbl 0962.35054 |
| [26] | Serrin, J.; Zou, H., Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9, 4, 635-653, (1996) · Zbl 0868.35032 |
| [27] | Serrin, J.; Zou, H., Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46, 369-380, (1998) · Zbl 0917.35031 |
| [28] | Souplet, P., The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221, 5, 1409-1427, (2009) · Zbl 1171.35035 |
| [29] | Souto, M. A.S., A priori estimates and existence of positive solutions of non-linear cooperative elliptic systems, Differential Integral Equations, 8, 1245-1258, (1995) · Zbl 0823.35064 |
| [30] | Wei, J.; Xu, X., Classification of solutions of higher order conformally invariant equations, Math. Ann., 313, 207-228, (1999) · Zbl 0940.35082 |
| [31] | Zou, H., A priori estimates for a semilinear elliptic systems without variational structure and their applications, Math. Ann., 323, 713-735, (2002) · Zbl 1005.35024 |
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