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Remarks about a generalized pseudo-relativistic Hartree equation. (English) Zbl 1433.35033

Based on the pioneering work of [V. Coti Zelati et al., Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 22, No. 1, 51–72 (2011; Zbl 1219.35292); S. Cingolani et al., Proc. R. Soc. Edinb., Sect. A, Math. 145, No. 1, 73–90 (2015; Zbl 1320.35300)], the authors study a class of generalized pseudo-relativistic Hartree equations \[ \left(-\Delta+m^2\right)^{\sigma}u+Vu=\left(W\ast F(u)\right)f(u) \] in \( \mathbb{R}^N\), where \(\sigma\in (0,1),\ m>0,\ V: \mathbb{R}^N\rightarrow \mathbb{R}\) is a continuous, (possibly) sign-changing bounded function satisfying suitable assumptions at infinity, \(W: \mathbb{R}^N\rightarrow \mathbb{R}\) is a radial convolution potential, and \(f: \mathbb{R}\rightarrow \mathbb{R}\) is a \(C^1\) nonlinearity with subcritical growth at infinity.
By means of the extension technique in [L. Caffarelli et al., Comm. Partial Differential Equ. 32, No. 8, 1245–1260 (2007)], they consider the Dirichlet-to-Neumann operator, and use the mountain pass lemma and the splitting lemma to prove the existence of a positive ground state solution. The authors also consider the regularity of the solution via cut-off, a bootstrap argument and the iteration process of Moser.

MSC:

35J20 Variational methods for second-order elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35R11 Fractional partial differential equations
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