Existence and Nonexistence for Semilinear Equations on Exterior Domains (Journal of Partial Differential Equations, 2020)
In this paper we prove the existence of an infinite number of radial solutions of $Δ u + K(r)f(u)= 0$ on the exterior of the ball of radius $R>0$ centered at the origin in ${\mathbb R}^{N}$ where $f$ is odd with $f0$ on $(β, δ),$ $f\equiv 0$ for $u> δ$, and where the function $K(r)$ is assumed to be positive and $K(r)→ 0$ as $r → ∞.$ The primitive $F(u) = \int_{0}^{u} f(s) \, {\rm d}s$ has a ``hilltop'' at $u=δ$ which allows one to use the shooting method to prove the existence of solutions.
Keywords: Semilinear; hilltop