## 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

### Keywords:

Hartree equation; Schrödinger equation
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