Block-radial symmetry breaking for ground states of biharmonic NLS

We prove that the biharmonic NLS equation
$\Delta^2u + 2\Delta u + (1 + ε)u = |u|^{p−2}u\ in\mathbb{R}^d$
has at least $k + 1$ geometrically distinct solutions if $ε > 0$ is small enough and $2 < p < 2^k_*$ , where $2^k_* $is an explicit critical exponent arising from the Fourier restriction theory of $O(d − k) × O(k)$-symmetric functions. This extends the recent symmetry breaking result of Lenzmann–Weth (Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates, 2023) and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of $k$. We further prove that, as $ε → 0^+$, the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces