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The proof of the Lane-Emden conjecture in four space dimensions. (English) Zbl 1171.35035
The author considers the following Lamé-Emden system:
\begin{aligned} & -\Delta u = v^p,\\ & - \Delta v = u^q,\end{aligned} in $$\mathbb R^n$$. The author proves that if $$n=3,4$$ and $$\frac{1}{p} + \frac{1}{q} > 1 - \frac{2}{n}$$, then the system above has no positive classical solutions. In the case $$n \geq 5$$ the author obtain a new region of nonexistence. The proof is based on Rellich-Pohozaev type identities, on a comparison property between components via the maximum principle, on Sobolev and interpolation inequalities on $$S^{n-1}$$ and on feedback and measure arguments.

##### MSC:
 35J45 Systems of elliptic equations, general (MSC2000) 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs 35B45 A priori estimates in context of PDEs 35J50 Variational methods for elliptic systems
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##### References:
 [1] Bidaut-Véron, M.-F.; Véron, L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. math., 106, 489-539, (1991) · Zbl 0755.35036 [2] Bidaut-Véron, M.-F.; Yarur, C., Semilinear elliptic equations and systems with measure data: existence and a priori estimates, Adv. differential equations, 7, 257-296, (2002) · Zbl 1223.35168 [3] Birindelli, I.; Mitidieri, E., Liouville theorems for elliptic inequalities and applications, Proc. roy. soc. Edinburgh sect. A, 128, 1217-1247, (1998) · Zbl 0919.35023 [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] Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke math. J., 63, 615-622, (1991) · Zbl 0768.35025 [6] 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 [7] de Figueiredo, D.G., Semilinear elliptic systems, (), 122-152 · Zbl 0955.35020 [8] 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 [9] 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 [10] de Figueiredo, D.G.; Yang, J., A priori bounds for positive solutions of a non-variational elliptic system, Comm. partial differential equations, 26, 2305-2321, (2001) · Zbl 0997.35015 [11] Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. pure appl. math., 34, 525-598, (1981) · Zbl 0465.35003 [12] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. partial differential equations, 6, 883-901, (1981) · Zbl 0462.35041 [13] 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 [14] Mitidieri, E., A Rellich type identity and applications, Comm. partial differential equations, 18, 125-151, (1993) · Zbl 0816.35027 [15] Mitidieri, E., Nonexistence of positive solutions of semilinear elliptic systems in $$\mathbb{R}^N$$, Differential integral equations, 9, 465-479, (1996) · Zbl 0848.35034 [16] Poláčik, P.; Quittner, P.; Souplet, Ph., Singularity and decay estimates in superlinear problems via Liouville-type theorems. part I: elliptic systems, Duke math. J., 139, 555-579, (2007) · Zbl 1146.35038 [17] Pucci, P.; Serrin, J., A general variational identity, Indiana univ. math. J., 35, 681-703, (1986) · Zbl 0625.35027 [18] Quittner, P.; Souplet, Ph., A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. ration. mech. anal., 174, 49-81, (2004) · Zbl 1113.35062 [19] Quittner, P.; Souplet, Ph., Superlinear parabolic problems. blow-up, global existence and steady states, Birkhäuser adv. texts, (2007), Springer Berlin · Zbl 1128.35003 [20] 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 [21] Serrin, J.; Zou, H., Non-existence of positive solutions of semilinear elliptic systems, (), 55-68 · Zbl 0900.35121 [22] Serrin, J.; Zou, H., Non-existence of positive solutions of Lane-Emden systems, Differential integral equations, 9, 635-653, (1996) · Zbl 0868.35032 [23] 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 [24] Souplet, Ph., Optimal regularity conditions for elliptic problems via $$L_\delta^p$$ spaces, Duke math. J., 127, 175-192, (2005) · Zbl 1130.35057 [25] 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 [26] 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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