×

Entire solutions of singular elliptic equations. (English) Zbl 0679.35003

This paper is concerned with the existence and asymptotic properties of solutions to the singular elliptic equation \[ -\Delta u=f(x)u^{- \lambda},\quad \lambda >0. \] It is assumed that f is positive and locally \(\alpha\)-Hölder continuous in \(R^ n\). The main results give sufficient conditions for the existence of positive, entire solutions having asymptotic behavior \[ (i)\quad u(x)\sim | x|^{2- n},\quad n\geq 3;\quad (ii)\quad u(x)\sim \log (x),\quad n=2. \] The method is based on the Schauder fixed point theorem and classical integral operator equations in \(R^ n\). It is of particular interest because it gives better information about rate of decay as \(| x| \to \infty\), and it generalizes to higher order equations where methods based on the maximum principle do not apply.
Reviewer: A.L.Edelson

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Berestycki, H.; Lions, P.L.; Peletier, L.A., An ODE approach to the existence of positive solutions for semilinear problems in Rn, Indiana univ. math. J., 30, No. 1, 141-157, (1981) · Zbl 0522.35036
[2] Callegari, A.J.; Nachman, A., A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. appl. math., 38, 275-281, (1980) · Zbl 0453.76002
[3] Callegari, A.J.; Nachman, A., Some singular, nonlinear differential equations arising in boundary layer theory, J. math. anal. apll., 64, 96-105, (1978) · Zbl 0386.34026
[4] {\scA. L. Edelson}, Semilinear elliptic equations in exterior domains, preprint. · Zbl 0646.35030
[5] {\scA. L. Edelson and J. D. Schuur}, Noniscillatory solutions of \((rx\^{}\{n\})\^{}\{n\} + ƒ(t, x) = 0\), Pacific J. Math., {\bf109}, No. 2, 313-325. · Zbl 0505.34027
[6] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second oder, () · Zbl 0691.35001
[7] {\scT. Kusano and M. Naito}, Nonlinear oscillation of fourth order differential equations, Canad. J. Math., {\bf28}, 80-852. · Zbl 0432.34022
[8] {\scT. Kusano and C. A. Swanson}, Asymptotic properties of semilinear elliptic equations, Funkcial. Ekvac., {\bf26}, No. 2, 115-129. · Zbl 0536.35024
[9] Kusano, T.; Swanson, C.A., Entire positive solutions of singular semilinear elliptic equations, Japan J. math., 11, No. 1, 145-155, (1985) · Zbl 0585.35034
[10] Noussair, E.S.; Swanso, C.A., Positive solutions of quasilinear elliptic equations in exterior domains, J. math. anal. appl., 75, 121-133, (1980) · Zbl 0452.35039
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.