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

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

2011-03-14

arxiv.org/abs/1103.2649v1

Physics

Mathematical Physics

Scientific paper

We prove compactness (up to translation) of minimizing sequences to: $$I_p^{\alpha,\beta}(\rho)=\inf_{\substack{u\in H^\frac 12(\R^3) \int_{\R^3}|u|^2 dx=\rho}} \frac{1}{2}\|u\|^2_{H^\frac 12(\R^3)} +\alpha\int\int_{\R^{3}\times\R^{3}} \frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-\beta\int_{\R^{3}}|u|^{p}dx$$ where $20$ and $\rho>0$ is small enough. In the case $p=\frac 83$ we show that similar compactness properties fail provided that in the energy above we replace the inhomogeneous Sobolev norm $\|u\|^2_{H^\frac 12(\R^3)}$ by the homogeneous one $\|u\|_{\dot H^\frac 12(\R^3)}$. We also provide a characterization of the parameters $\alpha, \beta>0$ in such a way that $I_\frac83^{\alpha,\beta}(\rho)>-\infty$ for every $\rho>0$.

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