Block-radial symmetry breaking for ground states of biharmonic NLS

We prove that the biharmonic NLS equation ${\mathrm{\Delta }}^{2}u+2\mathrm{\Delta }u+(1+\epsilon )u=\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ in ${\mathbb{R}}^d$ has at least $k+1$ different solutions if $\epsilon >0$ is small enough and $2<p<2_{\star }^k$, where $2_{\star }^k$ is an explicit critical exponent arising from the Fourier restriction theory of $O(d-k)\times O(k)$-symmetric functions. This extends the recent symmetry breaking result of Lenzmann-Weth and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of $k$. We further prove that, as $\epsilon \to 0^{+}$, the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces.