Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in \(\protect \mathbb{R}^N\). (English) Zbl 1490.35170
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 \(\Delta_1 u=\operatorname{div}(\tfrac{Du}{|Du|})\), \(V\) 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.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
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