×

A refined approach for non-negative entire solutions of \(\Delta u + u^p = 0\) with subcritical Sobolev growth. (English) Zbl 1375.35024

Summary: Let \(N\geq2\) and \(1<p<(N+2)/(N-2)_{+}\). Consider the Lane-Emden equation \(\Delta u+u^{p}=0\) in \(\mathbb{R}^{N}\) and recall the classical Liouville type theorem: if \(u\) is a non-negative classical solution of the Lane-Emden equation, then \(u\equiv0\). The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of J. Serrin and H. Zou [Differ. Integral Equ. 9, No. 4, 635–653 (1996; Zbl 0868.35032); Atti Semin. Mat. Fis. Univ. Modena 46, 369–380 (1998; Zbl 0917.35031)], originally used for the Lane-Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.

MSC:

35B08 Entire solutions to PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems, Discrete Contin. Dyn. Syst. 34 (2014), no. 9, 3317-3339. · Zbl 1304.35268
[2] J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana Univ. Math. J. 51 (2002), 37-51. · Zbl 1033.35032
[3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297. · Zbl 0702.35085
[4] C. C. Chen and C. S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes, Comm. Pure Appl. Math. 50 (1997), no. 10, 971-1017. · Zbl 0958.35013
[5] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615-622. · Zbl 0768.35025
[6] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1167-1184. · Zbl 1176.35067
[7] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst. 12 (2005), 347-354. · Zbl 1081.45003
[8] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330-343. · Zbl 1093.45001
[9] Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system, preprint (2014), .
[10] C. Cowan, A Liouville theorem for a fourth order Henon equation, Adv. Nonlinear Stud. 14 (2014), no. 3, 767-776. · Zbl 1301.35023
[11] D. G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa. Cl. Sci. (4) 21 (1994), no. 3, 387-397. · Zbl 0820.35042
[12] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of {\mathbb{R}^{n}}, J. Math. Pures Appl. 87 (2007), no. 5, 537-561. · Zbl 1143.35041
[13] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. · Zbl 0425.35020
[14] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in {{R}^{n}}, Adv. Math. Suppl. Studies A 7 (1981), 369-402.
[15] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883-901. · Zbl 0462.35041
[16] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525-598. · Zbl 0465.35003
[17] C. Gui, On positive entire solutions of the elliptic equation {Δ u+K(x)u^{p}=0} and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 2, 225-237. · Zbl 0861.35025
[18] G. Huang and C. Li, A Liouville theorem for high order degenerate elliptic equations, J. Differential Equations 258 (2015), no. 4, 1229-1251. · Zbl 1314.35055
[19] J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N. S.) 17 (1987), no. 1, 37-91. · Zbl 0633.53062
[20] Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst. 36 (2016), no. 6, 3277-3315. · Zbl 1336.35092
[21] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math. 123 (1996), 221-231. · Zbl 0849.35009
[22] Y. Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27-87. · Zbl 1173.35477
[23] Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), no. 2, 383-418.
[24] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in {{R}^{N}}, Differential Integral Equations 9 (1996), 465-480. · Zbl 0848.35034
[25] W. M. Ni, On the elliptic equation {Δ u+K(x)u^{(n+2)/(n-2)}=0}, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493-529. · Zbl 0496.35036
[26] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), 247-258. · Zbl 0236.53042
[27] Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differential Equations 252 (2012), 2544-2562. · Zbl 1233.35093
[28] P. Poláčik, Symmetry of nonnegative solutions of elliptic equations via a result of Serrin, Comm. Partial Differential Equations 36 (2011), no. 4, 657-669. · Zbl 1241.35083
[29] 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), no. 3, 555-579. · Zbl 1146.35038
[30] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Birkhäuser, Basel, 2007. · Zbl 1128.35003
[31] P. Quittner and P. Souplet, Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications, Comm. Math. Phys. 311 (2012), no. 1, 1-19. · Zbl 1254.35077
[32] W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations 161 (2000), no. 1, 219-243. · Zbl 0962.35054
[33] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 (1971), 304-318. · Zbl 0222.31007
[34] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations 9 (1996), no. 4, 635-653. · Zbl 0868.35032
[35] J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 369-380. · Zbl 0917.35031
[36] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math. 221 (2009), no. 5, 1409-1427. · Zbl 1171.35035
[37] P. Souplet, Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices, Netw. Heterog. Media 7 (2012), no. 4, 967-988. · Zbl 1263.35100
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.