The semirelativistic Choquard equation with a local nonlinear term

Bartosz Bieganowski; Simone Secchi

doi:10.3934/dcds.2019173

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2019, Volume 39, Issue 7: 4279-4302. Doi: 10.3934/dcds.2019173

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The semirelativistic Choquard equation with a local nonlinear term

Bartosz Bieganowski^1, , and
Simone Secchi^2,

1.
Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland
2.
Università degli Studi di Milano-Bicocca, Via R. Cozzi 55, I-20125 Milano, Italy

^* Corresponding author
^* Corresponding author

Received: October 31, 2018

Revised: December 31, 2018

Published: April 2019

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Abstract

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $ \mathbb{R}^N $

$ \begin{equation*} \sqrt{ -\Delta + m^2} u - mu + V(x)u = \left( \int_{ \mathbb{R} ^N} \frac{|u(y)|^p}{|x-y|^{N-\alpha}} \, dy \right) |u|^{p-2}u - \Gamma (x) |u|^{q-2}u, \end{equation*} $

where $ m > 0 $ and the potential $ V $ is decomposed as the sum of a $ \mathbb{Z}^N $-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space $ \mathbb{R}_{+}^{N+1} $.

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Mathematics Subject Classification: Primary: 35Q55, 35A15; Secondary: 35S05.

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