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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2412.14948 |
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| _version_ | 1866917874251071488 |
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| author | Silva, Edcarlos D. Oliveira, J. L. A. Goulart, C. |
| author_facet | Silva, Edcarlos D. Oliveira, J. L. A. Goulart, C. |
| contents | It is established existence of solution with prescribed $L^p$ norm for the following nonlocal elliptic problem:
\begin{equation*}
\left\{\begin{array}{cc}
\displaystyle (-Δ)^s_p u\ +\ V (x) |u|^{p-2}u\ = λ|u|^{p - 2}u + β\left|u\right|^{q-2}u\ \hbox{in}\ \mathbb{R}^N,
\displaystyle \|u\|_p^p = m^p,\ u \in W^{s, p}(\mathbb{R}^N).
\end{array}\right.
\end{equation*}
where $s \in (0,1), sp < N, β> 0 \text{ and } q \in (p, \overline{p}_s]$ where $\overline{p}_s =p+ sp^2/N$.
The main feature here is to consider $L^p$-subcritical and $L^p$-critical cases. Furthermore, we work with a huge class of potentials $V$ taking into account periodic potentials, asymptotically periodic potentials, and coercive potentials. More precisely, we ensure the existence of a solution of the prescribed norm for the periodic and asymptotically periodic potential $V$ in the $L^p$-subcritical regime. Furthermore, for the $L^p$ critical case, our main problem admits also a solution with a prescribed norm for each $β> 0$ small enough. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14948 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasilinear nonlocal elliptic problems with prescribed norm in the $L^p$-subcritical and $L^p$-critical growth Silva, Edcarlos D. Oliveira, J. L. A. Goulart, C. Analysis of PDEs It is established existence of solution with prescribed $L^p$ norm for the following nonlocal elliptic problem: \begin{equation*} \left\{\begin{array}{cc} \displaystyle (-Δ)^s_p u\ +\ V (x) |u|^{p-2}u\ = λ|u|^{p - 2}u + β\left|u\right|^{q-2}u\ \hbox{in}\ \mathbb{R}^N, \displaystyle \|u\|_p^p = m^p,\ u \in W^{s, p}(\mathbb{R}^N). \end{array}\right. \end{equation*} where $s \in (0,1), sp < N, β> 0 \text{ and } q \in (p, \overline{p}_s]$ where $\overline{p}_s =p+ sp^2/N$. The main feature here is to consider $L^p$-subcritical and $L^p$-critical cases. Furthermore, we work with a huge class of potentials $V$ taking into account periodic potentials, asymptotically periodic potentials, and coercive potentials. More precisely, we ensure the existence of a solution of the prescribed norm for the periodic and asymptotically periodic potential $V$ in the $L^p$-subcritical regime. Furthermore, for the $L^p$ critical case, our main problem admits also a solution with a prescribed norm for each $β> 0$ small enough. |
| title | Quasilinear nonlocal elliptic problems with prescribed norm in the $L^p$-subcritical and $L^p$-critical growth |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.14948 |