Fixed-L4 minimization in the harmonic-oscillator energy space
= Fixed-L4 minimization in the harmonic-oscillator energy space
{c}
{title2=$\|u\|_4^4=M$}
In two dimensions, the quadratic functional
$$
\int|\nabla u|^2+\int|u|^2+\frac\eta4\int|x|^2|u|^2
$$
attains its minimum subject to $\|u\|_4^4=M>0$. A bounded minimizing sequence is weakly compact in the harmonic-oscillator energy space and strongly compact in $L^4$: local <Rellich-Kondrachov compactness theorem> and the weighted tail bound prevent escape to infinity.