Block-radial symmetry breaking for ground states of biharmonic NLS

We prove that the biharmonic NLS equation $\Delta^2 u +2\Delta u+(1+\varepsilon)u=|u|^{p-2}u$ in $\mathbb R^d$ has at least $k+1$ different solutions if $\varepsilon>0$ 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 $\varepsilon\to 0^+$, the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces.


Introduction
A ground state for the biharmonic nonlinear Schrödinger equation, is a nonzero solution u ∈ H 2 (R d ) of (1.1), at which the infimum R ε (p) := inf , where q ε (u) := is attained.Lenzmann and Weth recently established the existence of nonradial ground states u ∈ H 2 (R d ) of (1.1).More precisely, [7,Theorem 1.2] guarantees the existence of a threshold ε 0 > 0 such that every ground state u ∈ H 2 (R d ) of (1.1) is a nonradial function if 0 < ε < ε 0 , as long as d ≥ 2 and 2 < p < 2 ⋆ .Here, 2 ⋆ := 2 d+1 d−1 is the endpoint Stein-Tomas exponent [14,15] from Fourier restriction theory.This symmetry breaking result is especially interesting in view of the recent work by Lenzmann and Sok [6] on sufficient conditions for radial symmetry of ground states in the general framework of elliptic pseudodifferential equations.Our goal is to shed further light on the interplay between symmetries and ground states of elliptic PDEs by generalizing the symmetry breaking result from [7] to a broader class of (pseudo-)differential equations and by including the symmetry groups Our main result implies that, for suitable exponents p > 2 and sufficiently small ε > 0, there exist radial and block-radial solutions of (1.1) whose energy is larger than that of the ground states.By our construction, this immediately implies that not only do ground states of (1.1) fail to be radial functions, but they also fail to be block-radial.The sharp Stein-Tomas exponent 2 k ⋆ for G k -symmetric functions from [12] will play a central role.It satisfies and is explicitly given by (1.2) We shall consider nonlinear elliptic (pseudo-)differential equations for a certain class of so-called (s, γ)-admissible symbols g ε .Here, g ε (|D|)u = F −1 (g ε (|•|)û).In the case s = γ = 2, the symbol g ε (|ξ|) = (|ξ| 2 − 1) 2 + ε of the biharmonic NLS is a prototypical example.
Definition 1.1.Let s > d d+1 and γ > 1.The symbol g ε : R → (0, ∞) is (s, γ)-admissible if it is measurable and satisfies the two-sided estimates for some constants 0 < c < C < ∞ independent of ε ∈ (0, 1).Some comments are in order.We assume γ > 1 in order to present a complete and unified analysis.Optimal results for γ ∈ (0, 1] require substantial modifications, which will not be investigated here.The assumption s > d/(d + 1) is equivalent to 2 ⋆ < 2d/(d − 2s) + , and thus ensures that the endpoint Stein-Tomas exponent is Sobolev-subcritical for the problem under investigation.In light of [2], this ensures the existence of ground states for (1.3), i.e., nonzero solutions which attain the infimum , where q ε (u) := For any fixed ε > 0, the functional q ε is positive and continuous on H s (R d ) as long as the symbol g ε is (s, γ)-admissible.More precisely, u → q ε (u) is then an equivalent norm on H s (R d ).Being interested in symmetric solutions of (1.3), we define the following radial and G k -symmetric versions of the Rayleigh quotient in (1.6): where the infima are taken over the Hilbert spaces identical minimizers up to a trivial change of coordinates.Hence in our main result we focus on k ∈ {1, . . ., ⌊d/2⌋}.
The strict inequalities in (1.7) rely on lower and upper bounds for the Rayleigh quotients that allow us to determine the asymptotic behaviour of R * ε (p) as ε → 0 + , for * ∈ {•, k, rad}.This is the content of our next result, Theorem 1.3.We introduce the interpolation parameter which satisfies α k ∈ (0, 1) if and only if 2 < p < 2 k ⋆ .We further define (1.9) which satisfies α rad ∈ (0, 1) if and only if 2 < p < 2 rad ⋆ .We also need the Sobolev exponent 1 (1.10) which ensures the largest possible range of validity in the next result.
s .Let g ε be (s, γ)-admissible for some s > d d+1 and γ > 1.As ε → 0 + , it holds that 1 We take this opportunity to record that the exponents defined in (1.2), (1.9) and (1.10) satisfy Here, A ε ∼ = B ε means that there exist constants 0 < c < C < ∞, independent of ε > 0, such that cB ε ≤ A ε ≤ CB ε holds for all sufficiently small ε > 0. We remark that the logarithmic term appearing when p = 2 rad ⋆ is new even in the special case of the biharmonic NLS, and thus refines the estimates from [7].This particular bound relies on a simple but nonstandard interpolation result that the interested reader may find in Appendix B.
The proof of Theorem 1.
for every G k -symmetric function f : R d → C, where the adjoint restriction (or extension) operator to the unit sphere More precisely, estimate (1.14) was shown in [12] for k ∈ {2, . . ., d − 2}, whereas the case k ∈ {1, d − 1} is a direct consequence of the classical Stein-Tomas inequality since That the range of exponents in (1.14) is optimal for L 2 -densities follows from [12,Theorem 1.3] and is based on a careful analysis of a G k -symmetric version of Knapp's well-known construction.Inequality (1.14) and these G k -symmetric Knapp-type constructions together pave the way towards the precise asymptotics given by Theorem 1.3.The asymptotics from Theorem 1.3 imply the chain of inequalities (1.7) stated in Theorem 1.2.To finish the proof of the latter, one still has to establish the existence of minimizers within the relevant class of functions.In turn, this amounts to a straightforward argument relying on certain compact embeddings of H s k , k ∈ {2, . . ., d − 2}, that we recall in Appendix A. If k ∈ {1, d − 1}, then the existence of a minimizer remains an open question, which we discuss in Appendix C.
Once the existence of minimizers has been established, their qualitative properties become of interest.To explore this matter, define the interval and the associated spherical shell This decomposition preserves radiality and G k -symmetry.Our next result reveals that u ε concentrates on the unit sphere in Fourier space, as ε → 0 We emphasize that this result also applies to the case * = k ∈ {1, d − 1} where the existence of a minimizer is still open.
In the smaller G k -symmetry breaking range 2 < p < 2 k ⋆ , minimizers become rough in Fourier space, as ε → 0 + .The intuition comes from the fact that the upper bounds in Theorem 1.3 were proved via test functions that resemble Knapp counterexamples.The Fourier transform of such functions vary sharply along spheres.Our next result indicates that such behaviour is somewhat necessary in order to be energetically efficient.The regularity is measured in the Sobolev space H t (S d−1 ) of functions having t ≥ 0 derivatives in L 2 (S d−1 ).This space is defined via spherical harmonic expansions, e.g. as in [9, §1.7.3, Remark 7.6], or equivalently by considering a smooth finite partition of unity and diffeomorphisms onto the unit ball of R d−1 together with the usual Sobolev norm on R d−1 .
, then for every t > 0 and every positive null sequence (δ ε ) ε>0 , it holds that Given that radially symmetric functions are constant on spheres around the origin, this phenomenon cannot occur in the radial case.
1.1.Structure of the paper.We prove Theorem 1.3 in §2, and this implies the first part of Theorem 1.2.We prove Theorem 1.4 in §3, and Theorem 1.5 in §4.We recall the classical proof of the second part of Theorem 1.2 in Appendix A and establish a technical interpolation result that is needed for the proof of Theorem 1.3 in Appendix B. Finally, Appendix C contains some further considerations regarding the exceptional cases k ∈ {1, d − 1}.
1.2.Notation.We use X Y or Y X to denote the estimate |X| ≤ CY for an absolute positive constant C, and X ∼ = Y to denote the estimates X Y X.We occasionally allow the implied constant C in the above notation to depend on additional parameters, which we will indicate by subscripts (unless explicitly omitted); thus for instance X j Y denotes an estimate of the form |X| ≤ C j Y for some C j depending on j.The indicator function of a set E is denoted by 1 E .

Lower and upper bounds
In this section, we prove Theorem 1.3.The following simple result will be useful for both lower and upper bounds.Lemma 2.1.Let ε ∈ (0, 1) and let g ε be (s, γ)-admissible for some s > d d+1 and γ > 1.
Proof.This follows from a simple change of variables and γ > 1: 2.1.Lower bounds.We start by addressing the lower bound in (1.12).We decompose an arbitrary and observe that v, w are still G k -symmetric.We then have u p ≤ v p + w p and q ε (u) = q ε (v) + q ε (w).It follows that Hence it suffices to prove lower bounds for the quotients corresponding to v and w separately.Since 2 < p < 2 ⋆ s , Sobolev embedding ensures

and thus
(2.2) where the ∼ =-estimate holds uniformly with respect to small ε > 0 thanks to (1.4).The lower bound for w −2 p q ε (w) follows at once, and so we focus on v.As in [7] the proof of lower bounds for v −2 p q ε (v) splits into two cases, according to whether or not p is larger than 2 k ⋆ .If p ≥ 2 k ⋆ , then we may use the G k -symmetric Stein-Tomas inequality (1.14) as follows: .
By Lemma 2.1, we then conclude which implies the claimed lower bound.If 2 < p < 2 k ⋆ , then the interpolation parameter from (1.8) satisfies α k ∈ (0, 1), and Plancherel's identity, the Fourier support assumption on v and the two-sided estimate (1.5) lead to From (2.3) with p = 2 k ⋆ and (2.4), it then follows that This concludes the proof of (1.12).The lower bounds in (1.11) are proved analogously given that it suffices to replace the exponent 2 k ⋆ originating from the G k -symmetric Stein-Tomas inequality by the exponent 2 ⋆ = 2 1  ⋆ coming from the classical Stein-Tomas inequality.We omit the details.(See [7, §4] for a proof in the special case s = γ = 2.) We now verify the lower bound in (1.13).Using the decomposition (2.1) and reasoning as in (2.2), it suffices to consider a radial function v ∈ H s rad whose Fourier support is contained in the spherical shell {ξ ∈ R d : Standard asymptotics for Bessel functions at zero and at infinity lead to the pointwise estimate where we used the Cauchy-Schwarz inequality, Lemma 2.1, and assumption (1.5) on g ε .Recall that 2 rad ⋆ = 2d d−1 .We thus infer (2.7) which implies the lower bound (1.13) in the range p > 2 rad ⋆ .If 2 < p < 2 rad ⋆ , then real interpolation between (2.7) and the simple 2 , (ε , as follows: if ε > 0 is sufficiently small, then This concludes the verification of the lower bounds in Theorem 1.3.

Upper bounds.
We first consider the upper bound in (1.12) for G k -symmetric functions.Given sufficiently small ε, δ > 0, define the G k -symmetric trial function v ε,δ via its Fourier transform as Here, both m ε,δ > 0 and the profile function a ε will be chosen below, and C k δ ⊂ S d−1 is a union of two spherical caps of radius δ defined as follows: Our estimates for v ε,δ make use of the following auxiliary result.Proposition 2.2.Let d ≥ 2 and k ∈ {1, . . ., d − 1}.There exist c 0 , c 1 , c 2 > 0 with the following property: for every (m, δ) ∈ (0, 1  2 ) 2 , there exist disjoint measurable sets whenever |r − 1| ≤ m and x ∈ E j .
Proof.If k ∈ {2, . . ., d − 2}, then this result is a variant of the computations from the proof of [12, Theorem 1.3], which we recall for the reader's convenience.The starting point is the formula for some sufficiently small absolute constant c > 0 satisfying c < 1 4 inf{z j+1 − z j : j ∈ N}.Here, {z j } j≥1 denotes the increasing sequence of local maxima of the Bessel function J (k−2)/2 , which satisfies z j = 2πj + O(1) as j → ∞.For every r ∈ [1 − m, 1 + m] and m ∈ (0, 1  2 ), we have E j ⊂ Ẽj (r), where To bound the measure of E j from below, choose α, β > 0 such that the estimates hold for (δ, m) ∈ (0, 1 2 ) 2 and all j ∈ N. We then find, for j = 1, . . ., This finishes the proof for k ∈ {2, . . ., d − 2}.Formula (2.9) continues to hold in the case k ∈ {1, d − 1} by taking into account the fact that The same proof, with z j replaced by 2πj, then yields the result.
Lemma 2.3.Assume 2 < p < 2k k−1 and let a ε ∈ L 2 loc (R + ) be nonnegative.Then, for all sufficiently small δ, ε, m ε,δ > 0 and v ε,δ as defined in (2.8), Proof.We alleviate notation by writing m = m ε,δ and v = v ε,δ .Estimate (2.10) is a simple consequence of first passing to polar coordinates in Fourier space, and then applying the two-sided inequality (1.5) together with the change of variables s = r − 1: Estimate (2.11) stems from the following identity, obtained via Fourier inversion: (2.12) Choosing c 0 > 0 and E j ⊂ R d as in Proposition 2.2 we obtain, for j ∈ {1, . . ., ⌊ c 0 δ 2 +m ⌋}, which is all we had to show.Note that we used p < 2k k−1 in the last estimate.
In the radial case, the following substitute holds for the simpler trial function given by Then, for all sufficiently small ε, m ε > 0 and v ε as defined in (2.14), (2.16) Proof.The proof is analogous to that of Lemma 2.3, so we just highlight the differences.Write m = m ε and v = v ε .The lower bound (2.16) relies on (2.6) for the representation Let again z j = 2πj + O(1), as j → ∞, denote the sequence of local maxima of J d−2

2
. With c > 0 as in the proof of Proposition 2.2, it holds that For such x, we get the pointwise lower bound Arguing as above, we find that this lower bound holds for x belonging to annular regions provided that j ∈ {1, 2, . . ., ⌊ c 0 m ⌋} for some suitably small c 0 > 0. Integrating these estimates and summing up the resulting bounds yields The three cases in (2.16) correspond to the exponent being larger than, equal to, or smaller than −1.
Proof of Theorem 1.3 (Upper bounds).Radial case (1.13).By Lemma 2.4, we have we find that the first factor simplifies to Choosing as claimed.Thus we may assume 2 < p ≤ 2 k ⋆ , which in particular forces p < 2k k−1 .So Lemma 2.3 applies and the choices a ε (s Non-symmetric case (1.11).In this case, we have This concludes the proof of Theorem 1.3.

Fourier concentration
In this section, we prove Theorem 1.4.
Let g ε be (s, γ)-admissible for some s > d d+1 and γ > 1.Let * ∈ {•, k, rad}.Assume that u ε is a minimizer for R * ε (p) and that there exist nonzero functions v ε , w ε , such that u ε = v ε + w ε and v ε • w ε = 0 almost everywhere.Moreover, suppose that there exists M ε > 1, such that Then the following estimates hold: This can be rewritten as which leads to the first estimate in (3.2) after elementary manipulations.The second estimate in (3.2) follows similarly: can be rewritten as which leads to the second estimate in (3.2).
Theorem 1.4 follows by verifying the lower bound (3.1) for some M ε such that M ε → ∞, as ε → 0 + .The latter condition is the content of our next result.

Roughness
In this section, we prove Theorem 1.5, which should be regarded as a quantified version of the fact that non-radial minimizers of the Rayleigh quotient asymptotically exhibit some sort of Knapp-type behaviour.In particular, their Fourier transforms must develop certain singularities along spheres which are near S d−1 .To prove this, we use Sobolev estimates for the adjoint restriction operator (1.15).In the general (non-symmetric) case, optimal results were established by Cho-Guo-Lee [3] but curiously their methods do not seem to allow for the required improved estimates in the G k -symmetric setting, which are key to our analysis.Proposition 4.1.Let d ≥ 2 and k ∈ {1, . . ., d − 1}.There exists t ⋆ = t ⋆ (d) > 0 such that, for all t ∈ [0, t ⋆ ) and G k -symmetric functions f ∈ H t (S d−1 ), the following estimate holds: , where Moreover, estimate (4.1) for k ∈ {1, d − 1} holds for every function f ∈ H t (S d−1 ).
Proof.In view of (1.14), the estimate holds for every G k -symmetric function f : S d−1 → C. On the other hand, it was proved in [11, Prop.1] that the pointwise decay estimate holds for every f ∈ C m (S d−1 ) with m = ⌊ d−1 2 ⌋ + 1.By Sobolev embedding, we may choose for every f ∈ H t ⋆ (S d−1 ).The desired conclusion follows from interpolating (4.2) and (4.3).Indeed, given t ∈ (0, t ⋆ ), define θ t := t/t ⋆ , so that real interpolation [1, Ch. 3] implies In the last equality, we used the identity Our aim is to use (4.4) in order to verify the hypothesis (3.1) in Lemma 3.1 with M ε → ∞, as ε → 0 + .To this end, split u ε = v ε + w ε as in (1.18), so that w ε is supported outside the spherical shell A ε,δε where δ ε → 0 + as ε → 0 + .In Lemma 3.2, we have already verified that this part is asymptotically negligible.More precisely, (4.5) with M ε → ∞, as ε → 0 + .Next we use (4.4) in order to prove lower bounds for the corresponding Rayleigh quotient involving v ε .To this end, we invoke Fourier inversion, the Sobolev estimate (4.1), assumption (4.4) and Lemma 2.1, yielding The key observation is that the exponent 2 k ⋆ (t) is strictly smaller than 2 k ⋆ , and therefore the corresponding interpolation parameter In view of (1.12), it follows that (4.6) From (4.5) and (4.6), we then conclude which is absurd since α k − α k (t) < 0 and M ε → ∞, as ε → 0 + .So the assumed uniform bound (4.4) cannot hold, and this completes the proof of the theorem.
can then be completed as in [2], yielding ũ as a minimizer for the Rayleigh quotient (C.
We now apply Proposition C.1 to the radially symmetric functions ) and spatial dimension d − 1 ≥ 2. This and the fractional Trace Theorem, i.e., the uniform boundedness of the trace operator H s (R d ) ∋ u → u| R d−1 ×{z} ∈ H t (R d−1 ) with respect to z ∈ R, implies, for some q > 2 and τ > 0, The last term is zero because {u n } is bounded in H s G and |x ′ n | → ∞ by assumption.This contradicts µ = 0, so {x ′ n } ⊂ R d−1 must be bounded.The preceding proof does not work for G 1 -symmetric functions since a priori the ũn are only G-symmetric.In other words, the ũn might not be even in the last coordinate.But if u denotes a G-symmetric minimizer/solution, then its reflection with respect to x d is another minimizer/solution.As a consequence, we find either one G 1 -symmetric minimizer or two distinct G-symmetric minimizers which are related to each other by a reflection.Given Theorem C.2 and Theorem 1.2, one may ask whether ground states are G-symmetric or not.Furthermore, as far as the asymptotic regime is concerned, one may check that the Fourier transform of any G-symmetric minimizer concentrates on the unit sphere and becomes rough as ε → 0 + ; the proof is identical to the G k -symmetric case.
3 naturally splits into two parts: lower and upper bounds for each of the Rayleigh quotients.The lower bounds hinge on the following recent G ksymmetric refinement of the Stein-Tomas inequality.If d ≥ 2 and k ∈ {1, . . ., d − 1}, then [12, Theorem 1.1] ensures the validity of the adjoint restriction inequality (1.14)

d 2
dr, where v(r) = v(rω) for r > 0 and almost every ω ∈ S d−1 .The appearance of the Bessel function is