Weighted Sobolev inequalities and superlinear elliptic problems on exterior domains

TL;DR

This paper establishes weighted Sobolev inequalities on exterior domains, proves compact embeddings, and applies these results to demonstrate existence, boundedness, and regularity of solutions for superlinear elliptic problems in unbounded regions.

Contribution

It introduces new weighted Sobolev inequalities on exterior domains and applies them to solve superlinear elliptic problems with boundary conditions.

Findings

Compact embedding of Beppo-Levi space into weighted Lebesgue spaces.

Existence of positive solutions for superlinear semipositone problems.

Solutions exhibit boundedness and regularity, with Green's function representations.

Abstract

Let $B_1 ^c = \{ x\in \mathbb{R}^N: |x|>1 \}, N \geq 2$, and $\mathcal{D}^{1,N}_0(B^c_1)$, be the Beppo-Levi space. We prove that $\mathcal{D}^{1,N}_0(B^c_1)$ is compactly embedded into the weighted Lebesgue space $L^r(B_1^c;K(x))$ for all $r\in[1,\infty)$ 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.