Uniform a priori bounds for Slightly Subcritical Elliptic Problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908399204040704 |
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| author | Cuesta, Mabel Pardo, Rosa |
| author_facet | Cuesta, Mabel Pardo, Rosa |
| contents | We obtain a uniform $L^{\infty}(Ω)$ a priori bound, for any positive weak solutions to elliptic problem with a nonlinearity $f$ slightly subcritical, slightly superlinear, and regularly varying.
To achieve our result, we first obtain a uniform estimate of an specific $L^1(Ω)$ weighted norm. This, combined with moving planes method and elliptic regularity theory, provides a uniform $L^\infty$ bound in a neighborhood of the boundary of $Ω$. Next, by using Pohozaev's identity, we obtain a uniform estimate of one weighted norm of the solutions. Joining now elliptic regularity theory, and Morrey's Theorem, we estimate from below the radius of a ball where a solution exceeds the half of its $L^\infty(Ω)$-norm. Finally, going back to the previous uniform weighted norm estimate, we conclude our result. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform a priori bounds for Slightly Subcritical Elliptic Problems Cuesta, Mabel Pardo, Rosa Analysis of PDEs 35B45, 35B09, 35B33 We obtain a uniform $L^{\infty}(Ω)$ a priori bound, for any positive weak solutions to elliptic problem with a nonlinearity $f$ slightly subcritical, slightly superlinear, and regularly varying. To achieve our result, we first obtain a uniform estimate of an specific $L^1(Ω)$ weighted norm. This, combined with moving planes method and elliptic regularity theory, provides a uniform $L^\infty$ bound in a neighborhood of the boundary of $Ω$. Next, by using Pohozaev's identity, we obtain a uniform estimate of one weighted norm of the solutions. Joining now elliptic regularity theory, and Morrey's Theorem, we estimate from below the radius of a ball where a solution exceeds the half of its $L^\infty(Ω)$-norm. Finally, going back to the previous uniform weighted norm estimate, we conclude our result. |
| title | Uniform a priori bounds for Slightly Subcritical Elliptic Problems |
| topic | Analysis of PDEs 35B45, 35B09, 35B33 |
| url | https://arxiv.org/abs/2506.07269 |