## 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
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### References:

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