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

Block-radial symmetry breaking for ground states of biharmonic NLS