Loitering at the hilltop on exterior domains

摘要

In this paper we prove the existence of an infinite number of radial solutions of $Delta u + f(u)= 0$ on the exterior of the ball of radius $R>0$ centered at the origin and $f$ is odd with $f 0$ on $(eta, delta),$ and $fequiv 0$ for $u> delta$. The primitive $F(u) = int_{0}^{u} f(t) , dt$ has a "hilltop" at $u=delta$ which allows one to use the shooting method and ODE techniques to prove the existence of solutions.