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Semiclassical analysis for pseudo-relativistic Hartree equations. (English) Zbl 1319.35204

Summary: In this paper we study the semiclassical limit for the pseudo-relativistic Hartree equation \[ \sqrt{-\varepsilon^2\Delta+m^2}u+Vu=(I_\alpha\ast |u|^p)|u|^{p-2}u,\quad \text{in }\mathbb R^N, \] where \(m>0\), \(2\leq p<\frac{2N}{N-1}\), \(V:\mathbb R^N \to\mathbb R\) is an external scalar potential, \(I_\alpha(x)=\frac{c_{N, \alpha}}{|x|^{N-\alpha}}\) is a convolution kernel, \(c_{N,\alpha}\) is a positive constant and \((N-1)p-N<\alpha<N\). For \(N=3\), \(\alpha=p=2\), our equation becomes the pseudo-relativistic Hartree equation with Coulomb kernel.

MSC:

35Q40 PDEs in connection with quantum mechanics
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