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The semirelativistic Choquard equation with a local nonlinear term

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  • 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} $.

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


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