Zhang, Zhijun A remark on the existence of entire solutions of a singular semilinear elliptic problem. (English) Zbl 0891.35042 J. Math. Anal. Appl. 215, No. 2, 579-582 (1997). Summary: It is proved that the singular semilinear elliptic equation \(-\Delta u= p(x)g(u)\), \(0\leq p(x)\), \(x\in\mathbb{R}^n\), \(2\leq n\), \(\lim_{s\to 0^+} g(s)=+\infty\), and \(g\in C^1((0,\infty), (0,\infty))\) which is strictly decreasing in \((0,\infty)\), has a unique positive \(C^{2+\alpha}_{\text{loc}}(\mathbb{R}^n)\) solution that decays to zero near \(\infty\) provided \(\int^\infty_0 t\varphi(t)dt< \infty\), where \(\varphi(t)= \max_{|x|= t}g(x)\). Cited in 24 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations Keywords:unique positive solution PDFBibTeX XMLCite \textit{Z. Zhang}, J. Math. Anal. Appl. 215, No. 2, 579--582 (1997; Zbl 0891.35042) Full Text: DOI References: [1] Lair, A. V.; Shaker, A. W., Entire solution of a singular semilinear elliptic problem, J. Math. Anal. Appl., 200, 498-505 (1996) · Zbl 0860.35030 [2] Wong, J. S., On the generalized Emden-Fowler equation, SIAM Rev., 17, 339-360 (1975) · Zbl 0295.34026 [3] Callegari, A. J.; Nachman, A., A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 30, 275-281 (1980) · Zbl 0453.76002 [4] Kusano, T.; Swanson, C. A., Entire positive solutions of singular semilinear elliptic equations, Japan. J. Math., 11, 145-155 (1985) · Zbl 0585.35034 [5] Dalmasso, R., Solutions dequations elliptiques semi-lineaires singulieres, Ann. Mat. Pura Appl., 153, 191-201 (1988) · Zbl 0692.35044 [6] Edelson, A., Entire solutions of singular elliptic equations, J. Math. Anal. Appl., 139, 523-532 (1989) · Zbl 0679.35003 [7] Shaker, A. W., On singular semilinear elliptic equations, J. Math. Anal. Appl., 173, 222-228 (1993) · Zbl 0785.35032 [8] Crandall, M. G.; Rabinowitz, P. H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2, 193-222 (1977) · Zbl 0362.35031 [9] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001 [10] Lazer, A. C.; McKenna, P. J., On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111, 721-730 (1991) · Zbl 0727.35057 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.