On a weighted semilinear Steklov problem in exterior domains

Abstract

Let B be the unit ball in $R^N$, $N\ge 3$. We are interested in a weighted elliptic problem in $R^N﹨\bar{B}$ with Steklov boundary conditions:(0.1)$\{\begin{array}{cc}-\text{div}\left(\right|x{|}^{\theta }\nabla u)=|x{|}^{\ell }{u}^p& \text{in }{R}^N﹨\bar{B},\\ u>0& \text{in }{R}^N﹨\bar{B},\\ \frac{\partial u}{\partial \nu }+du=0& \text{on ∂}\text{B},\end{array}$ with $d\in R$, $p>1$ and(0.2)$N^{\prime }:=N+\theta >2,\tau :=\ell -\theta >-2.$ A complete picture of existence and nonexistence of radial solutions for (0.1) is obtained. Furthermore, for $d<0$, the asymptotic behavior of radial solutions to (0.1) as $p\to \infty$ is studied.