Fixed-L4 minimization in the harmonic-oscillator energy space ($\Vert u{\Vert }_4^4=M$)

In two dimensions, the quadratic functionalattains its minimum subject to $\Vert u{\Vert }_4^4=M>0$. 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.