Weighted Sobolev inequalities and superlinear elliptic problems on exterior domains

Let $B_1^c=\{x\in {\mathbb{R}}^N:\mathrm{\mid }x\mathrm{\mid }>1\},N\ge 2$, and ${\mathcal{D}}_0^{1,N}(B_1^c)$, be the Beppo-Levi space. We prove that ${\mathcal{D}}_0^{1,N}(B_1^c)$ is compactly embedded into the weighted Lebesgue space $L^r(B_1^c;K(x))$ for all $r\in [1,\mathrm{\infty })$ for an appropriate class of weight functions $K$. As an application, we prove the existence of a positive solution to a superlinear semipositone problem on $B_1^c$ in ${\mathbb{R}}^2$. We also establish boundedness and regularity of solutions of certain boundary value problems and derive their Green’s function representation.