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

Cingolani, Silvia; Secchi, Simone

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.

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