Block-radial symmetry breaking for ground states of biharmonic NLS

Abstract

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