×

Qualitative properties of ground states for singular elliptic equations with weights. (English) Zbl 1115.35050

In this work, the authors deal with non-negative radial solutions of the singular quasilinear elliptic PDE \[ \text{div}(g(|x|)|\nabla u|^{m-2}\nabla u)+h(|x|)f(u)=0, \;x\in \mathbb R^n \setminus\{0\}, \] where \(g,h:\mathbb R^+\to \mathbb R^+\), \(f\in C(\mathbb R^+)\cap L^1(0,1)\), \(m>1\) and \(n\geq 1\). The singularities can appear in the functions \(g, h \) and \(f\) at the origin. This class contains (among other models) the generalized Matukuma equation.
The main theorems are devoted to establishing the uniqueness of ground states for such spatially dependent PDE under different conditions.

MSC:

35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Batt, Arch. Ration. Mech. Anal., 93, 159 (1986) · Zbl 0605.70008
[2] Chen, Commun. Partial Differ. Equations, 16, 1549 (1991)
[3] Clément, Asymptotic Anal., 17, 13 (1998)
[4] Coffman, Arch. Ration. Mech. Anal., 46, 81 (1972) · Zbl 0249.35029
[5] Cortázar, Adv. Differ. Equ., 1, 199 (1996)
[6] Cortázar, Arch. Ration. Mech. Anal., 142, 127 (1998) · Zbl 0912.35059
[7] Erbe, J. Differ. Equations, 138, 351 (1997) · Zbl 0884.34025
[8] Franchi, Adv. Math., 118, 177 (1996) · Zbl 0853.35035
[9] García-Huidobro, Adv. Differ. Equ., 6, 1517 (2001)
[10] Gazzola, Adv. Differ. Equ., 5, 1 (2000)
[11] Goncalves, Electron. J. Differ. Equ., 2004, 1 (2004)
[12] Kawano, J. Math. Soc. Japan, 45, 719 (1993)
[13] Kwong, M.K., Uniqueness of positive solutions for Δu-u+u^p=0 in ℝ^N. Arch. Ration. Mech. Anal. 105, 243-266 (1989) · Zbl 0676.35032
[14] Matukuma, T.: The cosmos. Tokyo: Iwanami Shoten 1938
[15] Mcleod, Trans. Am. Math. Soc., 339, 495 (1993)
[16] Mcleod, Arch. Ration. Mech. Anal., 99, 115 (1987)
[17] Montefusco, Adv. Differ. Equ., 6, 959 (2001)
[18] Peletier, Arch. Ration. Mech. Anal., 81, 181 (1983) · Zbl 0516.35031
[19] Peletier, J. Differ. Equations, 61, 380 (1986) · Zbl 0577.35035
[20] Pucci, Indiana Univ. Math. J., 47, 501 (1998)
[21] Pucci, Indiana Univ. Math. J., 47, 529 (1998)
[22] Pucci, J. Differ. Equations, 196, 1 (2004) · Zbl 1109.35022
[23] Serrin, Indiana Univ. Math. J., 49, 897 (2000) · Zbl 0979.35049
[24] Tso, J. Anal. Math., 52, 94 (1989)
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.