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Asymptotic behaviour of the solutions of sublinear elliptic equations with a potential. (English) Zbl 1034.35030

Summary: We study the limit behaviour of nonnegative solutions of the semilinear elliptic equation with a sublinear nonlinearity and a potential \[ -\Delta u-c\frac {c}{| x|^2}+ | x|^\sigma u^q=0\text{ in }\mathbb{R}^N\;(N\geq 3), \] where \(q\in(0,1)\), and \(c, \sigma \in\mathbb{R}\). The estimates lie upon a mean value inequality for the Helmholtz operator \(u\mapsto \Delta u+k^2u(k>0)\) in case \(c>0\). The behaviour of the solutions is essentially anisotropic, with possible dead cores.

MSC:

35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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[1] DOI: 10.1512/iumj.1983.32.32051 · Zbl 0548.35042 · doi:10.1512/iumj.1983.32.32051
[2] Berger M., Le spectre d’une variété riemannienne (1971) · doi:10.1007/BFb0064643
[3] DOI: 10.1007/BF01243922 · Zbl 0755.35036 · doi:10.1007/BF01243922
[4] Bidaut-VéroN M-F., Singularities in elliptic systems with absorption terms · Zbl 0533.35044
[5] Bidaut-Véron M-F., Singularities for a 2-dimensional semilinear elliptic equation with a non-Lipschitz linearity · Zbl 0533.35044
[6] DOI: 10.1080/03605309608821217 · Zbl 0932.35075 · doi:10.1080/03605309608821217
[7] rézis H., Arc. Rat. Mach. Anal 75 pp 1– (1980)
[8] Dautray R., Analyse Mathé et Calcul Numé, 2 Masson Ed. 1 (1987)
[9] Garnir H. G., Les Problé aux Limites de la physique Matématique (1958)
[10] DOI: 10.1002/cpa.3160340406 · Zbl 0465.35003 · doi:10.1002/cpa.3160340406
[11] Grichina G.V., On the compactness of the solutions of nonlinear elliptic and parabolic equations of second order in a lower bounded cylinder (in Russian)
[12] Grillot P., Thèse de Doctorat (1997)
[13] Guerch B., Rev. Mat. Iberoamericana 7 pp 65– (1991)
[14] DOI: 10.1080/03605308908820672 · Zbl 0697.35019 · doi:10.1080/03605308908820672
[15] John F., Partial Differential equations (1978) · Zbl 0426.35002 · doi:10.1007/978-1-4684-0059-5
[16] Licois J.R., A Clas of nonlinear conservative elliptic equations in cylinders · Zbl 0918.35051
[17] Olver F., Asymptotics and Special Functions (1953) · Zbl 0303.41035
[18] DOI: 10.1007/BFb0075139 · doi:10.1007/BFb0075139
[19] DOI: 10.1007/BF01811717 · Zbl 0467.35013 · doi:10.1007/BF01811717
[20] DOI: 10.1016/0362-546X(81)90028-6 · Zbl 0457.35031 · doi:10.1016/0362-546X(81)90028-6
[21] Yarur C., Singularidades de ecuaciones de Schrödinger estacionarias, Tesis doctoral (1984)
[22] Yarur C., Nonexistence of positive solutions for a class of semilinear elliptic systems · Zbl 0853.35038
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