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

Abstract

We 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.

Some recent results on singular p-laplacian systems

“…As far as we know, uniqueness of solutions to singular quasi-linear elliptic systems in the whole space is still an open problem. Taking inspiration from [13], a first result has been obtained by Gambera and Guarnotta [29].…”

Some recent results on singular $ p $-Laplacian systems

“…As far as we know, uniqueness of solutions to singular quasilinear elliptic systems in the whole space is still an open problem. Taking inspiration from [13], a first result has been obtained by Gambera and Guarnotta [29].…”

Some recent results on singular p-Laplacian equations

“…The arguments of both papers rely on a famous result by Diaz and Saa [84]. Theorem 1.3 of [85] contains a nice idea to achieve uniqueness for singular problems in exterior domains.…”