Weighted Sobolev inequalities and superlinear elliptic problems on exterior domains

TL;DR

The paper addresses compactness issues for the Beppo-Levi space $\mathcal{D}^{1,N}_0(B_1^c)$ in exterior domains at the critical case $p=N$ by proving a compact embedding into weighted spaces $L^r(B_1^c;K)$ under a decay $K(x)\le \frac{C}{|x|^{\gamma}}$ with $\gamma>N$. It develops a CK-N type interpolation inequality and uses density arguments to obtain continuous and compact embeddings for all $r\in[1,\infty)$, enabling variational methods on exterior domains. As an application in $\mathbb{R}^2$, it proves the existence of a positive solution to a superlinear semipositone problem on $B_1^c$ via the Mountain Pass theorem, together with regularity results and a Green’s function representation; Kelvin transform techniques yield boundedness and a priori control. Positivity of the obtained solution is established through Green’s representation and boundary-regularity arguments, highlighting the solution’s positivity throughout the exterior domain. Overall, the work extends weighted embedding theory and positivity results for exterior-domain elliptic problems, including nonradial weights, in the critical regime.

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.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1