TY - JOUR
T1 - Existence and Nonexistence for Semilinear Equations on Exterior Domains
AU - Iaia , Joseph A.
JO - Journal of Partial Differential Equations
VL - 4
SP - 299
EP - 316
PY - 2017
DA - 2017/11
SN - 30
DO - http://doi.org/10.4208/jpde.v30.n4.2
UR - https://global-sci.org/intro/article_detail/jpde/10676.html
KW - Semilinear
KW - hilltop
AB - <p>In this paper we prove the existence of an infinite number of radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius $R&gt;0$ centered at the origin in ${\mathbb R}^{N}$ where $f$ is odd with $f&lt;0$ on $(0, \beta) $, $f&gt;0$ on $(\beta, \delta),$&nbsp; $f\equiv 0$ for $u&gt; \delta$, and where the function $K(r)$ is assumed to be positive and $K(r)\to 0$ as $r \to \infty.$ The primitive $F(u)&nbsp; = \int_{0}^{u} f(s) \, {\rm d}s$ has a&nbsp; ``hilltop&#39;&#39; at $u=\delta$ which allows one to use the shooting method to prove the existence of solutions.</p>