Abstract
We consider the nonlinear Choquard equation
where \(N\ge 1\), \(I_\alpha \) is the Riesz potential integral operator of order \(\alpha \in (0, N)\) and \(p > 1\). If the potential \( V \in C (\mathbb {R}^N; [0,+\infty )) \) satisfies the confining condition
and \(\frac{1}{p} > \frac{N - 2}{N + \alpha }\), we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when \(p \ge 2\) the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. The growth assumption is sharp in view of a Pohožaev identity that we establish.
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Acknowledgements
Jean Van Schaftingen was supported by the Projet de Recherche (Fonds de la Recherche Scientifique–FNRS) T.1110.14 “Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations”. Jiankang Xia acknowledges the support of the NSF of China (NSFC-11271201), of the China Scholarship Council and the hospitality the Université catholique de Louvain (Institut de Recherche en Mathématique et en Physique).
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Van Schaftingen, J., Xia, J. Choquard equations under confining external potentials. Nonlinear Differ. Equ. Appl. 24, 1 (2017). https://doi.org/10.1007/s00030-016-0424-8
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DOI: https://doi.org/10.1007/s00030-016-0424-8
Keywords
- Nonlocal semilinear elliptic problem
- Weighted Sobolev embedding theorem
- Groundstate
- Fountain Theorem
- Least Action Nodal Solution