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Ground states for the pseudo-relativistic Hartree equation with external potential. (English) Zbl 1320.35300

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.

MSC:

35Q40 PDEs in connection with quantum mechanics
35B40 Asymptotic behavior of solutions to PDEs
35B09 Positive solutions to PDEs
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