Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in $\protect \mathbb{R}^N$
In this paper, we consider the following 1-Laplacian problem \[ -\Delta _1 u+V(x)\frac{u}{|u|}= f(x,u),\, x\in \mathbb{R}^N,\, u>0,\ u\in BV\left(\mathbb{R}^N\right), \] where $\Delta _1 u=\mathrm{div}(\tfrac{Du}{|Du|})$, $V$ is a periodic potential and $f$ is periodic and verifies the super-prim...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2022-04-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.276/ |
| Summary: | In this paper, we consider the following 1-Laplacian problem
\[ -\Delta _1 u+V(x)\frac{u}{|u|}= f(x,u),\, x\in \mathbb{R}^N,\, u>0,\ u\in BV\left(\mathbb{R}^N\right), \]
where $\Delta _1 u=\mathrm{div}(\tfrac{Du}{|Du|})$, $V$ is a periodic potential and $f$ is periodic and verifies the super-primary condition at infinity. By the Nehari type manifold method, the concentration compactness principle and some analysis techniques, we show the 1-Laplacian equation has a ground state solution. |
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| ISSN: | 1778-3569 |