Block-radial symmetry breaking for ground states of biharmonic NLS

We prove that the biharmonic NLS equation \(\begin{aligned} \Delta ^2 u +2\Delta u+(1+\varepsilon )u=|u|^{p-2}u\,\,\, in {\mathbb {R}}^d \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{2}u+2\mathrm{\Delta }u+(1+\epsilon )u={|u|}^{p-2}uin{\mathbb{R}}^d\end{array}$$ has at least \(k+1\) $k+1$ geometrically distinct solutions if \(\varepsilon >0\) $\epsilon >0$ is small enough and \(2<p<2_\star ^k\) $2<p<2_{\star }^k$ , where \(2_\star ^k\) $2_{\star }^k$ is an explicit critical exponent arising from the Fourier restriction theory of \(O(d-k)\times O(k)\) $O(d-k)\times O(k)$ -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 \(\varepsilon \rightarrow 0^+\) $\epsilon \to 0^{+}$ , the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces.