Weighted Sobolev inequalities and superlinear elliptic problems on exterior domains
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912672283361280 |
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| author | A, Ardra Ansari, Ameerraja Joseph, Anumol Sankar, Lakshmi |
| author_facet | A, Ardra Ansari, Ameerraja Joseph, Anumol Sankar, Lakshmi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted Sobolev inequalities and superlinear elliptic problems on exterior domains A, Ardra Ansari, Ameerraja Joseph, Anumol Sankar, Lakshmi Analysis of PDEs 35J20, 35B09, 35J08 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. |
| title | Weighted Sobolev inequalities and superlinear elliptic problems on exterior domains |
| topic | Analysis of PDEs 35J20, 35B09, 35J08 |
| url | https://arxiv.org/abs/2510.23157 |