Existence of positive radial solutions for some quasilinear elliptic equations in annular domains

Abstract

The existence of positive radial solutions of the equation-div(|Du| p−2 Du)=f(u) is studied in annular domains in R n, n≥2. It is proved that if f(0)≥0, f is somewhere negative in (0, ∞), lim u→0+f′(u)=0 and lim u→∞(f(u)/u p−1)=∞, then there is a large positive radial solution on all annuli. If f(0)<0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.