Ground states for semi-relativistic Schrödinger-Poisson-Slater energies

Abstract

We prove the existence of ground states for the semi-relativistic Schrödinger-Poisson-Slater energy $$I^{\alpha ,\beta }(\rho )=\underset{\begin{array}{c}u\in H^{\frac{1}{2}}({\mathbb{R}}^3){\int }_{{\mathbb{R}}^3}|u{|}^{2}dx=\rho \end{array}}{\inf }\frac{1}{2}\Vert u{\Vert }_{H^{\frac{1}{2}}({\mathbb{R}}^3)}^2+\alpha \int {\int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3}\frac{|u(x){|}^2|u(y){|}^2}{|x-y|}dxdy-\beta {\int }_{{\mathbb{R}}^3}|u{|}^{\frac{8}{3}}dx$$ $\alpha ,\beta >0$ and $\rho >0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $\alpha ,\beta >0$ such that $I^{\alpha ,\beta }(\rho )>-\infty$ for every $\rho >0$, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant $S>0$ such that $$\frac{1}{S}\frac{\Vert \phi {\Vert }_{L^{\frac{8}{3}}({\mathbb{R}}^3)}}{\Vert \phi {\Vert }_{{\dot{H}}^{\frac{1}{2}}({\mathbb{R}}^3)}^{\frac{1}{2}}}\le {\left(\int {\int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3}\frac{|\phi (x){|}^2|\phi (y){|}^2}{|x-y|}dxdy\right)}^{\frac{1}{8}}$$ for all $\phi \in C_0^{\infty }({\mathbb{R}}^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $\Vert u{\Vert }_{H^{\frac{1}{2}}({\mathbb{R}}^3)}^2$ by the homogeneous one $\Vert u{\Vert }_{{\dot{H}}^{\frac{1}{2}}({\mathbb{R}}^3)}$.

17 pages

References in corpus (4)

• Boson Stars as Solitary Waves
• Stable standing waves for a class of nonlinear Schroedinger-Poisson equations
• Minimizers for the Hartree-Fock-Bogoliubov Theory of Neutron Stars and White Dwarfs
• Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations

Cited by in corpus (5)

• Ground state solutions for non-autonomous fractional Choquard equations
• Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity
• On fractional Choquard equations
• Sharp lower bounds for Coulomb energy
• Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems