Cingolani, Silvia; Secchi, Simone Ground states for the pseudo-relativistic Hartree equation with external potential. (English) Zbl 1320.35300 Proc. R. Soc. Edinb., Sect. A, Math. 145, No. 1, 73-90 (2015). Summary: We prove the existence of positive ground state solutions to the pseudo-relativistic Schrödinger equation \[ \sqrt{-\Delta+m^2} u+Vu= (W*|u|^\theta)|u|^{\theta-2} u\quad\text{in }\mathbb{R}^N,\quad u\in H^{1/2}(\mathbb{R}^N), \] where \(N\geq 3\), \(m>0\), \(V\) is a bounded external scalar potential and \(W\) is a radially symmetric convolution potential satisfying suitable assumptions. We also provide some asymptotic decay estimates of the found solutions. Cited in 1 ReviewCited in 24 Documents MSC: 35Q40 PDEs in connection with quantum mechanics 35B40 Asymptotic behavior of solutions to PDEs 35B09 Positive solutions to PDEs Keywords:Hartree equation; Schrödinger equation PDF BibTeX XML Cite \textit{S. Cingolani} and \textit{S. Secchi}, Proc. R. Soc. Edinb., Sect. A, Math. 145, No. 1, 73--90 (2015; Zbl 1320.35300) Full Text: DOI arXiv OpenURL