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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1902.04578 |
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| _version_ | 1866914970647658496 |
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| author | Della Pietra, Francesco Piscitelli, Gianpaolo |
| author_facet | Della Pietra, Francesco Piscitelli, Gianpaolo |
| contents | Let us consider the following minimum problem \[ λ_α(p,r)= \min_{\substack{u\in W_{0}^{1,p}(-1,1)\\ u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1}|u'|^{p}dx+α\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\displaystyle\int_{-1}^{1}|u|^{p}dx}, \] where $α\in\mathbb R$, $p\ge 2$ and $\frac p2 \le r \le p$. We show that there exists a critical value $α_C=α_C (p,r)$ such that the minimizers have constant sign up to $α=α_{C}$ and then they are odd when $α>α_{C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1902_04578 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems Della Pietra, Francesco Piscitelli, Gianpaolo Analysis of PDEs Let us consider the following minimum problem \[ λ_α(p,r)= \min_{\substack{u\in W_{0}^{1,p}(-1,1)\\ u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1}|u'|^{p}dx+α\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\displaystyle\int_{-1}^{1}|u|^{p}dx}, \] where $α\in\mathbb R$, $p\ge 2$ and $\frac p2 \le r \le p$. We show that there exists a critical value $α_C=α_C (p,r)$ such that the minimizers have constant sign up to $α=α_{C}$ and then they are odd when $α>α_{C}$. |
| title | Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1902.04578 |