Analysis of positive solutions for classes of quasilinear singular problems on exterior domains

Abstract

AbstractWe consider the problem\left\{\begin{aligned} \displaystyle{-}\Delta_{p}u&\displaystyle=K(x)\frac{f(u% )}{u^{\delta}}&&\displaystyle\text{in }\Omega^{e},\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\\ \displaystyle u(x)&\displaystyle\to 0&&\displaystyle\text{as }|x|\to\infty,% \end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}} ({N>2}) is a simply connected bounded domain containing the origin with {C^{2}} boundary {\partial\Omega}, {\Omega^{e}:=\mathbb{R}^{N}\setminus\overline{\Omega}} is the exterior domain, {1<p<N} and {0\leq\delta<1}. In particular, under an appropriate decay assumption on the weight function K at infinity and a growth restriction on the nonlinearity f, we establish the existence of a positive weak solution {u\in C^{1}(\overline{\Omega^{e}})} with {u=0} pointwise on {\partial\Omega}. Further, under an additional assumption on f, we conclude that our solution is unique. Consequently, when Ω is a ball in {\mathbb{R}^{N}}, for certain classes of {K(x)=K(|x|)}, we observe that our solution must also be radial.