Standing waves with a critical frequency for nonlinear Choquard equations

Van Schaftingen, Jean;Xia, Jiankang
(2017) Nonlinear Analysis — Vol. 161, p. 87-107 (2017)

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Abstract
In this paper, we study the nonlocal Choquard equation −ε2Δuε+Vuε=(Iα∗|uε|p)|uε|p−2uε where N≥1, Iα is the Riesz potential of order α∈(0,N) and ε>0 is a parameter. When the nonnegative potential V∈C(ℝN) achieves 0 with a homogeneous behaviour or on the closure of an open set but remains bounded away from 0 at infinity, we show the existence of groundstate solutions for small ε>0 and exhibit the concentration behaviour as ε→0.
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Van Schaftingen, J., & Xia, J. (2017). Standing waves with a critical frequency for nonlinear Choquard equations. Nonlinear Analysis, 161, 87-107. https://doi.org/10.1016/j.na.2017.05.014 (Original work published 2017)