Abstract
In this work we prove the existence of ground state solutions for the following class of problems
where \(\lambda > 0\), \(\Delta _1\) denotes the 1-Laplacian operator which is formally defined by \(\Delta _1 u = \text{ div }(\nabla u/|\nabla u|)\), \(V:\mathbb {R}^N \rightarrow \mathbb {R}\) is a potential satisfying some conditions and \(f:\mathbb {R} \rightarrow \mathbb {R}\) is a subcritical nonlinearity. We prove that for \(\lambda > 0\) large enough there exist ground-state solutions and, as \(\lambda \rightarrow +\infty \), such solutions converges to a ground-state solution of the limit problem in \(\Omega = \text{ int }( V^{-1}(\{0\}))\).
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Acknowledgements
The authors would like to thank the anonymous referee for careful reading the manuscript and by drawing their attention to crucial references on the 1-Laplacian operator. C.O.Alves was partially supported by CNPq/Brazil 304804/2017-7. G. M. Figueiredo is supported by CNPq and FAPDF. M.T.O. Pimenta is supported by FAPESP 2019/14330-9 and CNPq 303788/2018-6.
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Alves, C.O., Figueiredo, G.M. & Pimenta, M.T.O. Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in \(\mathbb {R}^N\). Bull Braz Math Soc, New Series 51, 863–886 (2020). https://doi.org/10.1007/s00574-019-00179-4
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DOI: https://doi.org/10.1007/s00574-019-00179-4