Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Jimmy Petean; Juan Carlos Fern\'andez; Oscar Palmas

REVIEW 5 major objections 4 minor 44 references

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

T0 review · 5 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A Yamabe-type equation on the round sphere admits sign-changing solutions with arbitrarily many prescribed isoparametric nodal components, even for supercritical exponents.

desk verdict Theorem 1.1's p-range overreaches: the proof only supports the exponent from the smaller focal submanifold, but the statement uses the larger one; the rest of the paper is solid enough to warrant review. read the letter →

arxiv 1908.08091 v1 pith:BZ7TD7OE submitted 2019-08-21 math.AP

Juan Carlos Fernández , Jimmy Petean , Oscar Palmas This is my paper · ORCID

classification math.AP MSC 34B1635B0635B3335B4453C2158E2058J05

keywords nodalsolutionssupercriticalellipticequationsYamabeproblemisoparametrichypersurfacessingularODEdoubleshootingmethodharmonicmorphismsblow-upanalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that a semilinear elliptic equation on the round sphere has infinite families of sign-changing solutions, including when the exponent is supercritical. The solutions are built as functions of an isoparametric function, so their level sets are isoparametric hypersurfaces and their singular behavior concentrates on the focal submanifolds. The proof reduces the PDE to a singular second-order ODE and then uses a double shooting argument, showing that two phase-plane curves spiral outward in opposite directions and therefore must intersect. This yields, for every positive integer $k$, a nodal solution with at least $k$ disjoint isoparametric nodal components, with the solutions blowing up on a focal submanifold as $k$ grows. The same reduction, pushed through harmonic morphisms, gives sequences of sign-changing solutions to the Yamabe problem on complex and quaternionic projective spaces.

What carries the argument

The central object is the isoparametric function $f:\mathbb{S}^n\to[-1,1]$, a smooth function whose gradient norm and Laplacian depend only on $f$; its regular level sets are isoparametric hypersurfaces and its two singular levels $M_-$ and $M_+$ are focal submanifolds. Writing $u=w\circ f$ transforms the PDE into the singular ODE $w''+\frac{h(r)}{\sin r}w'+\frac{\lambda}{\ell^2}(|w|^{p-1}w-w)=0$ on $[0,\pi]$, where $h$ is strictly decreasing with a single zero. The proof then uses double shooting from the two singular endpoints: two phase-plane curves $I(d)$ and $J(c)$ are defined by evolving with $w'(0)=0$ and $w'(\pi)=0$. An imported oscillation theorem forces the number of zeroes near each endpoint to diverge as $|d|,|c|\to\infty$, while a Pohozaev-type identity shows the curves have unbounded radius; the resulting opposite-oriented spirals must intersect, producing solutions with arbitrarily many zeroes and critical points.

What would settle it

Numerically integrate the singular ODE (2.4) for the case $m_-=m_+=2$, take $p$ close to the endpoint $(m_++3)/(m_+-1)=5$, and count the zeroes of $w_d$ in a small interval $(0,\varepsilon)$ as $d$ grows; if the zero count does not diverge to infinity, the oscillation theorem's range is not valid and the double-shooting intersections cannot be forced.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the equation $-\Delta_{g_0}u+\lambda u=\lambda|u|^{p-1}u$ on $\mathbb{S}^n$, with $\lambda>0$ and $p>1$, admits a nodal solution $u_k$ for every $k\in\mathbb{N}$ whenever $p$ lies in $(1,\frac{n-n_++2}{n-n_+-2})$, a range that contains the supercritical regime above the Sobolev critical exponent. The nodal set of $u_k$ contains at least $k$ connected components, each an isoparametric hypersurface diffeomorphic to a prescribed one, while the critical set contains the two focal submanifolds and at least $k-1$ further isoparametric hypersurfaces. Moreover, $|u_k|\to\infty$ on one of the focal submanifolds as $k\to\infty$, so the sequence is not compact. The projective-space results transfer these sphere solutions to sign-changing Yamabe solutions on $\mathbb{CP}^m$ and $\mathbb{HP}^m$ with isoparametric hypersurfaces as level sets and unbounded sup-norms.

Load-bearing premise

The load-bearing premise is the imported oscillation assertion for the singular initial-value problem: for $H(0)>0$ and $p$ in the claimed range, solutions with large initial value $d$ have arbitrarily many zeroes arbitrarily close to $0$; the paper's printed inequality (2.5) appears to need to read $(H(0)+1)/2 < (p+1)/(p-1)$ to match the $p$-interval used later, and if that corrected criterion fails at the boundary values, the double-shooting intersection argument collapses.

Editorial extensions

If this is right

For every isoparametric hypersurface $S$ and every integer $k$, the sphere equation admits a nodal solution with at least $k$ nodal components, each diffeomorphic to $S$.
The constructed solutions blow up on one focal submanifold: $|u_k(x)|\to\infty$ for every $x\in M_-$ or every $x\in M_+$ as $k\to\infty$, so the solution family is noncompact.
In the critical Yamabe case on the sphere, when both focal submanifolds have positive dimension, there is a sign-changing sequence tending to $+\infty$ on one focal submanifold and to $-\infty$ on the other.
On $\mathbb{CP}^m$ and $\mathbb{HP}^m$ with their canonical metrics, the Yamabe equation has sign-changing solutions for every $k$, with level sets given by isoparametric hypersurfaces and sup-norms diverging to infinity.
The admissible $p$-interval extends beyond the critical exponent $(n+2)/(n-2)$, so supercritical nodal solutions are not merely possible but appear in infinite families.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the argument does not use the full symmetry group of the round metric, only the existence of an isoparametric function with two focal submanifolds; analogous nodal families should exist on other manifolds carrying such functions, provided the corresponding singular ODE has the same oscillation and energy properties.
Editorial inference: the blow-up along minimal focal submanifolds is reminiscent of known concentration phenomena, but here it arises from ordinary differential equations and shooting rather than Lyapunov-Schmidt reduction, suggesting these solutions are explicit models for submanifold concentration.
Editorial inference: the harmonic-morphism transfer is likely not limited to $\mathbb{CP}^m$ and $\mathbb{HP}^m$; any compact Riemannian submersion with minimal fibers over a base admitting an invariant isoparametric function should produce analogous unbounded nodal solutions on the base.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read the paper and largely agree with the conditional verdict, though I want to be sharper about the p-range. The stress-test note is right: Theorem 1.1 overreaches. The proof fixes 1 ≤ m_- ≤ m_+ and derives the admissible range p < (m_+ + 3)/(m_+ - 1), which is the exponent set by the larger multiplicity, i.e., the smaller focal submanifold. But the statement orders n_- ≤ n_+ and puts n_+ in the denominator. Since n_+ is then the larger focal dimension, the printed range can be infinite. For S^1 × S^3 in S^5 the theorem claims p < ∞, while the proof gives p < 3. That is a real gap, not a typo. Theorem 1.4 has the same problem if read under the proof's labeling. The fix is straightforward: use min(n_-, n_+) in the exponent, or reverse the ordering so n_+ is the smaller dimension.

Credit where due: the double-shooting strategy and the energy appendix are substantial. Lemma A.2's Pohozaev-type identity is a real addition, and the transfer to CP^m and HP^m through harmonic morphisms is a nice trick. Corollary 1.3 explicitly uses κ = min(dim M_-, dim M_+), so it survives the p-range correction.

The rest of the soft spots are textual but should be fixed: inequality (2.5) is misprinted; it has to be (H(0)+1)/2 < (p+1)/(p-1), otherwise the algebra in (2.6) doesn't follow. Lemma 3.2(2) has wrong dimensions: α = β = 4 cannot occur for S^{15}, and M_+ is written as S^{β-1} instead of S^{4β-1}. The last paragraph of Theorem 1.1's proof swaps M_- and M_+: arccos(f) = 0 at M_+, not M_-. None of these should sink the paper, but they need cleaning up.

The oscillating-solution theorem is imported from the authors' earlier work; that is not a flaw, but it means the paper's genuinely new contribution is the energy analysis and the projective-space reduction, not the zero-counting engine.

Who is this for? Researchers in Yamabe-type equations and isoparametric hypersurfaces. It deserves a serious referee and, with the p-range fixed, should be publishable. My recommendation: send it to review, flag the p-range as a required major revision, and check the Lemma 3.2 numerology.

Referee Report

5 major / 4 minor

Summary. The paper studies the supercritical and subcritical semilinear elliptic problem −Δu + λu = λ|u|^{p−1}u on the round sphere S^n, restricted to functions of the form u = w∘f for an isoparametric function f. The reduction turns the PDE into a singular ODE on [0,π] whose coefficient h has one zero. The authors use a double shooting method from the two singular endpoints, together with an oscillation theorem for the associated initial value problem, to produce sign-changing solutions with arbitrarily many nodal components that are isoparametric hypersurfaces, and with blow-up on one of the focal submanifolds. The paper also derives a Yamabe-type multiplicity result on complex and quaternionic projective spaces via harmonic morphisms. The main claimed theorem (Theorem 1.1) is an infinite nodal solution result in the supercritical range.

Significance. If the main theorem is correct after the necessary corrections, it would be a substantial advance: it would provide infinitely many sign-changing solutions to the Yamabe-type problem in a genuinely supercritical range, with a precise description of the level and critical sets as isoparametric hypersurfaces and focal submanifolds, and with blow-up at focal submanifolds. The double-shooting/energy method in Appendix A is substantial, and the reduction via isoparametric functions and harmonic morphisms is elegant. The paper contains no fitted parameters, and the geometric constructions are explicit. However, the central claim as stated is currently overreaching: the p-range in Theorem 1.1 is not the range proved in Section 2, and several auxiliary statements contain dimension or label errors that must be corrected before the result is trustworthy.

major comments (5)

[Theorem 1.1 and Section 2, Eq. (2.6)] Theorem 1.1 states the p-range with n_+ while assuming n_- ≤ n_+, but the proof in Section 2 fixes 1 ≤ m_- ≤ m_+ and derives (2.6) with n_+ denoting the smaller focal dimension. Under the theorem's ordering, n_+ is the larger focal dimension, so the stated range is strictly larger than the range proved. For example, for the Clifford hypersurface S^1×S^3 in S^5, the theorem would allow every p > 1, whereas the proof's admissible range is 1 < p < 3. The double-shooting intersection argument therefore cannot justify Theorem 1.1 as stated. The theorem should either be restricted to p < (n−n_-+2)/(n−n_-−2) with n_- the smaller focal dimension, or the proof should be reworked to handle both focal dimensions symmetrically. The same mismatch appears in Theorem 1.4.
[Section 2, Theorem 2.1 and Eq. (2.5)] The inequality printed in (2.5), H(0) + 1/2 < (p+1)/(p−1), does not imply the later range (2.6). Substituting H(0)=m would give p < (2m+3)/(2m−1), which is not the bound used. The algebra that follows (2.6) is consistent with the intended criterion (H(0)+1)/2 < (p+1)/(p−1), which is equivalent to p < (m+3)/(m−1). Equation (2.5) should be corrected to that form; as printed, the statement of the imported oscillation theorem is internally inconsistent with its use.
[Section 3, Lemma 3.2(2) and its proof] The proof of part (2) contains dimension errors. The lemma claims m ≥ 3, but the proof begins 'For each m ≥ 4'; for m=3 it sets α=β=4, which would give a real dimension 32 for H^4×H^4, while S^{4m+3}=S^15 requires ambient dimension 16. The expression for the ambient space as H^α × H^β × R^{2m+2} also has the wrong real dimension, and the focal submanifold formula writes M_+ = {0}×S^{β−1} rather than {0}×S^{4β−1}. These inconsistencies undermine the construction of the SU(2)-invariant Cartan–Münzner polynomial needed for Corollary 1.3. The construction is repairable by taking α+β = m+1 with α=β=2 for m=3, but as written the proof is not correct.
[Section 3, proof of Corollary 1.2] The final sentence of the proof of Corollary 1.2 says 'Since arctan(f(x))=0 for every x∈M_+ and arctan(f(x))=π for every x∈M_-', which appears to have the focal labels swapped, and the expression u_k = arctan(f(w_k)) does not make sense because f is a function on S^n while w_k is a scalar. The intended construction is presumably u_k = z_k∘f with z_k = arctan(w_k). This affects the signs in the blow-up limits claimed in Corollary 1.2 and needs to be corrected.
[Section 2, Lemma 2.3] The scaling in Lemma 2.3 is hard to parse and contains inconsistent exponents. The displayed definition of z_d prints d^{-2/(p-1)} w_{d^{2/(p-1)}}(r/(d√λ)), while the inversion formula reads w_d(r)=d z_{d^{(p-1)/2}}(√λ d^{(p-1)/2} r); these expressions are not equivalent as written. Since the lemma is used to prove that the zeroes r_j(d) collapse to 0, which is essential for the spiral argument, the proof should be rewritten with coherent scaling exponents.

minor comments (4)

[Abstract and Introduction] There are several typos, e.g. 'hipersurfaces' in the abstract and 'integerdivide{0}' artifacts that should read '\setminus\{0\}'. A proofreading pass is needed.
[Section 2, after Lemma 2.4] The notation ρ(d) is reused for both the phase-plane radial variable ρ(r,d) and the endpoint value ρ(d,a0); please use a different symbol for one of them to avoid confusion.
[Section 3, Lemma 3.2(1)] In part (1), for k=0 the text says α=β=2 if k=1 and α=k, β=3 if k≥2, but the case k=0 (i.e. m=2) is not covered even though the lemma states m ≥ 3; this is likely a harmless omission, but it should be clarified.
[Appendix A, Step 3] The sentence 'the reader may skip it in a first reading' is informal for a journal article and should be removed or rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the ODE reduction and double-shooting argument are genuine derivations, and the imported oscillation theorems are independent general results with explicit hypotheses.

full rationale

I found no step in which a claimed prediction or first-principles result reduces by construction to its own inputs. The reduction u = w ∘ f is an equivalence between the PDE and the singular ODE (1.9), not a definition of the desired solution, and the double-shooting construction uses the curves I and J to produce intersections rather than assuming the target nodal solution exists. The principal imported ingredient, Theorem 2.1, is a general oscillation criterion for the singular initial value problem (2.4) with explicit hypotheses H(0) > 0, p > 1 and inequality (2.5); its conclusion concerns zeroes of solutions near 0, not the existence of nodal solutions to the sphere problem, so it is not a restatement of Theorem 1.1. It is cited from [22,27], and while [22] shares authors with the present paper, the cited result is parameter-free, externally checkable, and does not include the target result in its assumptions, so under the stated rules it counts as independent support. The zero-counting formulas (2.8) and the phase-plane angle behavior are also imported from [22], but they are elementary consequences of the setup and do not smuggle in the final multiplicity conclusion. The unboundedness of the shooting spirals is proved in this paper via the energy and Pohozaev-type estimates in Appendix A. There is a separate consistency issue between the p-ranges in the statements and the proof convention 1 ≤ m_- ≤ m_+, but that is a correctness or indexing concern, not a circularity of the kinds enumerated. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the chosen solution. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; lambda, p, ell, m_- and m_+ are inputs or geometric constants from prior literature. The central claim rests on the cited oscillation theorem and the standard theory of isoparametric hypersurfaces.

assumptions (5)

domain assumption Oscillation theorem for singular ODE (Theorem 2.1): under H(0)>0, p>1 and the stated inequality, solutions of (2.4) have arbitrarily many zeroes near the singularity.
Imported from [22,27] and not proved in the paper; it is the main engine for producing many zeroes from large initial values. Location: Section 2, Theorem 2.1.
standard math Classification of isoparametric hypersurfaces on spheres: number of principal curvatures ell in {1,2,3,4,6}, with multiplicities m_- and m_+ satisfying the relations used in (1.9).
Used to write a(t), b(t), h(r) explicitly in Section 1 and to derive the p-range (2.6).
standard math Existence and uniqueness for the singular initial value problem (2.4).
Used throughout Section 2 and Lemma 2.2; cited from [22,29].
domain assumption Hopf fibrations S^{2m+1}->CP^m and S^{4m+3}->HP^m are Riemannian submersions with minimal fibers, hence harmonic morphisms.
Used in Section 3 to transfer solutions from spheres to projective spaces; standard from [3,24].
standard math Muenzer relation n = 1 + ell(m_- + m_+)/2 and dimension formulas for focal submanifolds.
Used in Lemma 3.2 and in translating the p-range in Corollary 1.3.

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Cite this review

Pith. "Pith review of Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces." pith.science (2026). https://pith.science/paper/BZ7TD7OE

@misc{pith2026190808091,
  author       = {Pith},
  title        = {Pith review of: Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BZ7TD7OE}},
  note         = {Machine review of arXiv:1908.08091}
}

abstract

Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -\Delta_{g_0}u+\lambda u=\lambda\ | u\ | ^{p-1}u\qquad\text{ on }\mathbb{S}^n \] with $\lambda>0$ and $1<p$, into a singular ODE in $[0,\pi]$ of the form $w'' + \frac{h(r)}{\sin r} w' + \frac{\lambda}{\ell^2}\ (| w|^{p-1}w - w\ )=0$, where $h$ is an strictly decreasing function having exactly one zero in this interval and $\ell$ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to $f$ and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when $p>\frac{n+2}{n-2}$, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

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