On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations
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| Format: | Preprint |
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2022
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| _version_ | 1866915857953718272 |
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| author | Bobkov, Vladimir Tanaka, Mieko |
| author_facet | Bobkov, Vladimir Tanaka, Mieko |
| contents | We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-Δ_p u = λm(x)|u|^{p-2}u + ηa(x)|u|^{q-2}u + f(x)$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $q<p$. Under certain regularity and qualitative assumptions on the weights $m, a$ and the source function $f$, we identify ranges of parameters $λ$ and $η$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $η=0$, and among them there are results providing new information even in the linear case $p=2$. In particular, we show that for any $p>1$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the $p$-Laplacian provided $m,f \in L^γ(Ω)$ with $γ>N$. For completeness, we also investigate the existence of solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_08898 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations Bobkov, Vladimir Tanaka, Mieko Analysis of PDEs Classical Analysis and ODEs 35J92, 35B50, 35B65, 35B09, 35B30, 35A01, 35B38 We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $-Δ_p u = λm(x)|u|^{p-2}u + ηa(x)|u|^{q-2}u + f(x)$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $q<p$. Under certain regularity and qualitative assumptions on the weights $m, a$ and the source function $f$, we identify ranges of parameters $λ$ and $η$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $η=0$, and among them there are results providing new information even in the linear case $p=2$. In particular, we show that for any $p>1$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the $p$-Laplacian provided $m,f \in L^γ(Ω)$ with $γ>N$. For completeness, we also investigate the existence of solutions. |
| title | On the antimaximum principle for the $p$-Laplacian and its sublinear perturbations |
| topic | Analysis of PDEs Classical Analysis and ODEs 35J92, 35B50, 35B65, 35B09, 35B30, 35A01, 35B38 |
| url | https://arxiv.org/abs/2210.08898 |