We study the radial symmetry of minimizers to the Schr\"odinger-Poisson-Slater (S-P-S) energy: $$\inf_{\substack{u\in H^1(\R^3)\\ \|u\|_{L^2(\R^3)}=\rho}} \frac 12 \int_{\R^3} |\nabla u|^2 + \frac 14 \int_{\R^3} \int_{\R^3} \frac{|u(x)|^2|u(y)|^2}{|x-y|}dxdy - \frac 1p \int_{\R^3} |u|^p dx$$ provided that $2<p<3$ and $\rho$ is small. The main result shows that minimizers are radially symmetric modulo suitable translation.
On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy
PRINARI, Francesca Agnese;
2012
Abstract
We study the radial symmetry of minimizers to the Schr\"odinger-Poisson-Slater (S-P-S) energy: $$\inf_{\substack{u\in H^1(\R^3)\\ \|u\|_{L^2(\R^3)}=\rho}} \frac 12 \int_{\R^3} |\nabla u|^2 + \frac 14 \int_{\R^3} \int_{\R^3} \frac{|u(x)|^2|u(y)|^2}{|x-y|}dxdy - \frac 1p \int_{\R^3} |u|^p dx$$ provided that $2File in questo prodotto:
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