Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms

Jin, Tianling

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 6, pp. 965-981.

We prove some symmetry property for equations with Hardy terms in cones, without any assumption at infinity. We also show symmetry property and nonexistence of entire solutions of some elliptic systems with Hardy weights.

Classification : 35B06, 35B08, 35B09
Mots clés : Hardy–Sobolev inequality, Caffarelli–Kohn–Nirenberg inequality, Lane–Emden system, Nonexistence

[1] T. Aubin, Problemes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 no. 4 (1976), 573-598 | MR | Zbl

[2] T. Bartsch, S. Peng, Z. Zhang, Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities, Calc. Var. Partial Differential Equations 30 no. 1 (2007), 113-136 | MR | Zbl

[3] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bull. Braz. Math. Soc. 22 no. 1 (1991), 1-37 | MR | Zbl

[4] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 no. 3 (1984), 259-275 | EuDML | Numdam | MR | Zbl

[5] M. Calanchi, B. Ruf, Radial and non radial solutions for Hardy–Hénon type elliptic systems, Calc. Var. Partial Differential Equations 38 no. 1 (2010), 111-133 | MR | Zbl

[6] D. Cao, Y.Y. Li, Results on positive solutions of elliptic equations with a critical Hardy–Sobolev operator, Methods Appl. Anal. 15 no. 1 (2008), 81-96 | MR

[7] F. Catrina, Z.Q. Wang, On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math. 54 no. 2 (2001), 229-258 | MR | Zbl

[8] J.L. Chern, C.S. Lin, Minimizers of Caffarelli–Kohn–Nirenberg inequalities with the singularity on the boundary, Arch. Ration. Mech. Anal. 197 no. 2 (2010), 401-432 | MR | Zbl

[9] D.G. De Figueiredo, P.L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa 21 (1994), 387-397 | EuDML | Numdam | MR | Zbl

[10] D.G. De Figueiredo, I. Peral, J.D. Rossi, The critical hyperbola for a Hamiltonian elliptic system with weights, Ann. Mat. Pura Appl. 187 no. 3 (2008), 531-545 | MR | Zbl

[11] H. Egnell, Positive solutions of semilinear equations in cones, Trans. Amer. Math. Soc. 330 no. 1 (1992), 191-201 | MR | Zbl

[12] N. Ghoussoub, X.S. Kang, Hardy–Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 no. 6 (2004), 767-793 | EuDML | MR | Zbl

[13] N. Ghoussoub, F. Robert, The effect of curvature on the best constant in the Hardy–Sobolev inequalities, Geom. Funct. Anal. 16 no. 6 (2006), 1201-1245 | MR | Zbl

[14] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 no. 12 (2000), 5703-5743 | MR | Zbl

[15] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 no. 3 (1979), 209-243 | MR | Zbl

[16] V. Glaser, A. Martin, H. Grosse, W. Thirring, A family of optimal conditions for the absence of bound states in a potential, E.H. Lieb, B. Simon, A.S. Wightman (ed.), Studies in Mathematical Physics, Princeton University Press (1976), 169-194

[17] C.H. Hsia, C.S. Lin, H. Wadade, Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal. 259 no. 7 (2010), 1816-1849 | MR | Zbl

[18] Y.Y. Li, C.S. Lin, A nonlinear elliptic PDE with two Sobolev–Hardy critical exponents, preprint, 2010.

[19] Y.Y. Li, L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 no. 1 (2003), 27-87 | MR | Zbl

[20] Y.Y. Li, M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 no. 2 (1995), 383-418 | MR

[21] E.H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. 118 no. 2 (1983), 349-374 | MR | Zbl

[22] F. Liu, J. Yang, Nontrivial solutions of Hardy–Henon type elliptic systems, Acta Math. Sci. Ser. B Engl. Ed. 27 no. 4 (2007), 673-688 | MR | Zbl

[23] E. Mitidieri, Non-existence of positive solutions of semilinear systems in ${ℝ}^n$, Differential Integral Equations 9 no. 3 (1996), 465-479 | MR | Zbl

[24] R. Musina, Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal. 68 no. 12 (2008), 3972-3986 | MR | Zbl

[25] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 no. 4 (1971), 304-318 | MR | Zbl

[26] J. Serrin, H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differential Integral Equations 9 no. 4 (1996), 635-653 | MR | Zbl

[27] J. Serrin, H. Zou, Existence of positive entire solutions of elliptic Hamiltonian systems, Comm. Partial Differential Equations 23 no. 3–4 (1998), 577-599 | MR | Zbl

[28] J. Serrin, H. Zou, Existence of positive solutions of the Lane–Emden system, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 369-380 | MR | Zbl

[29] E.M. Stein, G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 no. 4 (1958), 503-514 | MR | Zbl

[30] M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer Verlag (2008) | MR | Zbl

[31] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353-372 | MR | Zbl