{"id":"f60a7f40-081f-43a7-8d10-a1d08502be72","arxiv_id":"2411.14686","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper classifies existence, uniqueness, and multiplicity for the supercritical Lane-Emden equation on cones with inhomogeneous Dirichlet data, under the exponent condition H(p-1)<0.","lead":"This paper analyzes the Lane-Emden equation on an infinite cone with a prescribed boundary value, and determines exactly when positive solutions exist depending on the size of that boundary value. It proves that small boundary data always admit a solution, large data never do, and in a specific exponent range there is exactly one solution at the critical threshold and at least two just below it.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's critical criterion H(p−1)<0 is derived from the sharp conical Hardy constant A=((N−2)/2)^2+Λ in Lemma 5.3; that geometric inequality is cited to [25] and not proved, so if its constant is off the whole classification collapses.","rationale":"The reader identified the same external input, and I agree. I checked the surrounding algebra: the equivalence between (5.32) and H(p−1)<0 is correct, being a direct expansion with q=p−1, and the use of the Hardy inequality in Lemma 5.3 is otherwise as stated. I also looked for internal gaps—the deferred stability Proposition 4.1, the compactness of G in Lemma 6.1, and the C^1 regularity of the map Φ—and found no demonstrated inconsistency; those points are either standard or recoverable from the cited arguments. The novel, geometry-dependent threshold is the one place where a single wrong constant would invalidate the theorem's central claim. Because the concern is checkable and external rather than a proven error, the reader's CONDITIONAL assessment is the appropriate verdict, with no change needed.","tokens_in":26674,"tokens_out":29764,"duration_ms":296300,"concrete_test":"Verify [25, Proposition 4.1] independently by spherical-harmonic reduction: for u(r,θ)=f(r)φ_1(θ), where φ_1 is the first Dirichlet eigenfunction of −Δ' on A with eigenvalue Λ, compute the quotient ∫|∇u|² / ∫u²/r² and show its infimum over f∈C_c^∞(0,∞) equals ((N−2)/2)^2+Λ, approached by f_R(r)=η(r/R)r^{-(N−2)/2} as R→∞. If the infimum is below A, then (1.16) is false and the proof fails; if it equals A, the external input is confirmed. Separately expand a general u in the eigenbasis of −Δ' on A to check that no angular mode lowers the constant below A.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The existence of a solution at κ=κ* and the subsequent uniqueness and multiplicity statements rest on the coercivity estimate (5.21) in Lemma 5.3. The proof of Lemma 5.3 uses the conical Hardy inequality (1.16) at exactly one place: (5.30) replaces ∫ψ²|x|^{2τ−2}w_κ² by A^{-1}∫|∇(ψ|x|^τ w_κ)|² with A=((N−2)/2)^2+Λ. That coefficient, together with the 1/p term, forces the condition (5.20), and after choosing τ=2/(p−1)−(N−2)/2 it becomes (5.32), which Proposition 5.1 identifies with H(p−1)<0. The inequality (1.16) is not proved in this paper; it is imported from [25, Proposition 4.1]. Proposition 2.1 and the Perron construction in Section 2 do not establish it. Thus a single external geometric fact about the cone's first Laplace–Beltrami eigenvalue Λ carries the theorem's new threshold. If the constant in [25] were smaller than A, or if the inequality failed for some D^{1,2}_0 functions on the cone, Lemma 5.3 would not yield (5.21), the uniform bound (5.33) in Lemma 5.4 would not follow, and Proposition 5.1 would have no solution at κ=κ*; Theorem 1.2(i)–(iii) would lose its support. This is exactly the step the paper advertises as the new ingredient, namely the difference of the constant in the Hardy inequality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the supercritical Lane–Emden equation -Δu = u^p on an infinite cone Ω with inhomogeneous Dirichlet data κμ on ∂Ω\\{0}. The main results are Theorem 1.1, which gives a threshold κ* such that minimal solutions exist for 0<κ<κ* and no solutions exist for κ>κ*, and Theorem 1.2, which under p<p_JL and H(p-1)<0 (with H the cubic in (1.12) depending on N and the cone eigenvalue Λ) adds existence of a unique solution at κ=κ* and the existence of at least two solutions for κ in an interval (κ*,κ*). The proofs combine a Perron/supersolution construction, linearized eigenvalue problems and stability of minimal solutions, new weighted energy estimates based on the sharp conical Hardy inequality (1.16), and a Crandall–Rabinowitz bifurcation argument. Section 7 states analogous results for Hénon-type equations. The algebraic passage from (5.32) to H(p-1)<0 is correct, and the condition H(p-1)<0 is derived rather than imposed.","tokens_in":26961,"tokens_out":30081,"duration_ms":269951,"significance":"If the results are correct, the paper gives a rather complete existence/nonexistence and multiplicity classification for a supercritical boundary-value problem on cones, extending to the critical boundary case and to multiplicity earlier work on bounded domains, the half-space, and the forced Lane–Emden equation. The main novelty is a set of weighted energy estimates whose coercivity condition is linked to the constant ((N-2)/2)^2+Λ in the sharp Hardy inequality on cones; this produces a genuinely domain-dependent cubic condition H(p-1)<0. The proof structure is coherent, and the paper is careful to distinguish what is proved from what is imported from the literature. In particular, no free parameters are fitted and the threshold condition is not assumed ad hoc. The main caveat is that the pivotal estimate (5.21) relies on the cited sharp Hardy inequality (1.16), so the authors should make the applicability of that inequality explicit; with that clarification, the argument is convincing.","major_comments":[],"minor_comments":[{"comment":"The displayed inequalities 'α−2>pα and β−2<pβ' have the wrong direction. From (1.8) one has α−2<pα and β−2>pβ, which are precisely the inequalities needed for the lower bound −ΔV_{α,β}≥CU_{pα,pβ}. Please correct the direction of these inequalities.","section":"Section 3, proof of Proposition 3.1"},{"comment":"In the second inequality of (5.23), the term τ²|x|^{2τ−2}(wκ)^2 should carry the cutoff ψ², i.e. it should read τ²ψ²|x|^{2τ−2}(wκ)^2. Without ψ², the subsequent application of the Hardy inequality (5.30) to ψ|x|^τ wκ is not justified, and the displayed bound is not what is used later.","section":"Section 5, Lemma 5.3, Eq. (5.23)"},{"comment":"The sentence 'Combining (5.23)–(5.26), (5.30), and (5.31)' appears to contain a wrong cross-reference: (5.31) is the conclusion of the lemma rather than an input. It should refer to (5.29) in place of (5.31).","section":"Section 5, Lemma 5.3, proof"},{"comment":"The text 'We find τ1 and τ2 satisfying (5.2)' should refer to condition (5.20) (or the displayed system (5.34)) rather than to (5.2), which is a different condition in Lemma 5.1.","section":"Section 5, Lemma 5.4, proof"},{"comment":"In (5.17) and (5.18), the exponent inside the integral appears to be 2qν/(p−1), not 2q/((p−1)ν), if the displayed estimate is to follow from Lemma 5.1 with the outer exponent (p−1)/(2qν). Please correct the exponent and verify the surrounding algebra.","section":"Section 5, Lemma 5.1, Eqs. (5.17)–(5.18)"},{"comment":"Please state explicitly that ψ|x|^τ wκ belongs to D^{1,2}_0(Ω), for example via Lemma 4.4(iii) and the compact support of ψ away from the vertex and infinity, so that the conical Hardy inequality (1.16) applies. It would also be helpful to quote the precise statement of [25, Proposition 4.1] to confirm that the constant A=((N-2)/2)^2+Λ is exactly the one used.","section":"Section 5, Lemma 5.3, Eq. (5.30)"},{"comment":"The symbol vκ is introduced in Section 4 as vκ=vκ_{j*}, but in the estimates around (5.7) the relevant quantity appears to be vκ_{j*+1}. Please reconcile the notation or explicitly state which index is used in Lemma 5.1 and in (5.26).","section":"Section 4 vs. Section 5"},{"comment":"The statement of the Crandall–Rabinowitz theorem includes both Fκ(u*,κ*)∉Im(Fu) and Fκ(u*,κ*)∉Z, where Z is merely a complementary subspace of the kernel. In the standard theorem these are not independent hypotheses; the correct assumption is non-membership in the image. Since in the application Z is taken to be the image, please rephrase the proposition to avoid this ambiguity.","section":"Proposition 6.1"},{"comment":"The opening sentence 'Assume the same conditions as in Theorem 1.1' should presumably read 'Theorem 7.1' when stating the Hénon-type generalization.","section":"Section 7, Theorem 7.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound in its overall strategy, and the stress-test concern about the single external Hardy inequality does not, on careful reading, amount to a fatal gap: (1.16) is a standard sharp inequality and the test functions are in D^{1,2}_0 once the simple verification is added. The main issues are typographical and presentational, but they are concentrated in the pivotal Section 5, so the author should correct them carefully. I would be happy to see a revised version rather quickly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is real: a complete existence/nonexistence/uniqueness/multiplicity picture for the supercritical Lane-Emden problem with inhomogeneous Dirichlet data on cones, under the condition H(p-1)<0. That cubic condition depends on the cone's first eigenvalue, so the geometry of the domain genuinely enters the critical exponent. I checked the algebra linking the weighted energy condition (5.32) to H(p-1)<0; it is correct. The main technical novelty, the weighted estimates in Lemma 5.3 that exploit the sharper conical Hardy constant A=((N-2)/2)^2+Lambda, is presented clearly and does the job it advertises.\n\nThe paper is also honest: it states the open cases p>=p_JL and H(p-1)>=0, and there are no fitted parameters or target-amounted assumptions. The existence, stability, and bifurcation parts hang together coherently.\n\nSoft spots are present but minor. The load-bearing step is the conical Hardy inequality (1.16), cited to Nazarov, which sets the constant that later becomes H(p-1)<0. The paper does not prove it, so a referee should confirm that the constant is indeed sharp and the inequality valid on the cone. That is a normal citation to a published result, not a flaw, but it is the one external input on which the threshold rests. Also, several key steps (stability of minimal solutions, properties of the first eigenpair) are deferred to previous papers by the same circle, so independent verification requires pulling those arguments. That is standard practice, but it does slow a referee down.\n\nOverall, the proof structure is coherent, the new exponent condition is derived rather than imposed, and the main theorem is a genuine step beyond the half-space and whole-space results. The reader's take matches my reading: the concern about the Hardy constant is worth checking, but on the evidence here it does not land as a fatal flaw.\n\nThis paper is for anyone working on nonlinear elliptic equations in unbounded domains, Liouville-type exponents, or bifurcation for boundary value problems. I would send it to a serious referee without hesitation.","headline":"Solid, genuinely new classification result for the Lane-Emden equation on cones; the domain-dependent exponent condition is real, and the paper deserves a serious referee.","tokens_in":27544,"tokens_out":1556,"would_cite":true,"duration_ms":17077,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J25","35J61","35B09","35B32"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single cone eigenvalue sets the threshold structure of supercritical Lane–Emden solutions.","keywords":["Lane–Emden equation","supercritical nonlinearity","infinite cone","inhomogeneous Dirichlet boundary condition","Joseph–Lundgren exponent","multiple positive solutions","Hardy inequality","bifurcation theory"],"falsifier":"Fix a cone with cross-section $A$ where $H(p-1)<0$ and $p<p_{JL}$, and numerically continue the branch of positive solutions of $(L_\\kappa)$ as $\\kappa$ increases: the theorem predicts a minimal branch up to $\\kappa_*$, a second branch for $\\kappa_*<\\kappa<\\kappa^*$, a single solution at $\\kappa^*$, and none above it. A branch count different from that — for instance, no solution at $\\kappa^*$, three solutions in the interval, or a solution for $\\kappa>\\kappa^*$ — would refute the classification. Separately, checking the best constant in (1.16) for that cone decides whether the external Hardy-inequality input really is sharp.","tokens_in":26394,"feed_emoji":"📐","tokens_out":18797,"duration_ms":172219,"temperature":0.7,"pith_summary":"This paper studies the Lane–Emden equation $-\\Delta u=u^p$ on an infinite cone in $\\mathbb{R}^N$, with a prescribed boundary value $\\kappa\\mu$ on the cone's boundary away from the tip. It asks how the size $\\kappa$ of the boundary data orders the set of positive solutions, and answers with a classification in the supercritical range $p>p^*_\\gamma$, $p<p_{JL}$, and $H(p-1)<0$: two thresholds $0\\le\\kappa_*<\\kappa^*$ exist such that a minimal solution exists for $0<\\kappa\\le\\kappa_*$, a unique solution exists at $\\kappa=\\kappa^*$, no solution exists for $\\kappa>\\kappa^*$, and at least two solutions exist for $\\kappa_*<\\kappa<\\kappa^*$. The decisive object is the cubic $H(q)$ of (1.12), whose coefficients involve only the dimension $N$ and the first eigenvalue $\\Lambda$ of the cone's cross-section, so the shape of the domain enters the boundary-data dichotomy through a single spectral quantity. The paper also proves a threshold-existence theorem (Theorem 1.1) valid for every $p>p^*_\\gamma$, and extends both results to Hénon-type nonlinearities $K(x)u^p$ with $K(x)\\sim |x|^a$.","feed_headline":"One cone eigenvalue sets the boundary-data solution thresholds","feed_subtitle":"A cubic in that eigenvalue controls whether the equation has minimal, unique, or multiple positive solutions.","key_machinery":"Three mechanisms together carry the argument. The weighted space $C_{\\alpha,\\beta}$ with weight $U_{\\alpha,\\beta}(x)=|x|^\\alpha(1+|x|^2)^{(\\beta-\\alpha)/2}$ encodes admissible growth at the tip and decay at infinity, and the nonlinear problem is rewritten as a fixed-point equation $\\Phi(u,\\kappa)=0$ with $\\Phi$ a smooth map on this Banach space. The sharp conical Hardy inequality $((N-2)/2)^2+\\Lambda)\\int_\\Omega |x|^{-2}\\varphi^2\\le \\int_\\Omega |\\nabla\\varphi|^2$ supplies the coercivity behind the new weighted energy estimates (Lemmas 5.3 and 5.4), and a direct calculation converts the resulting condition (5.32) into the cubic inequality $H(p-1)<0$. Finally, the linearized operator at the extremal solution has one-dimensional kernel spanned by the first eigenfunction $\\varphi_{\\kappa^*}$, and a standard bifurcation theorem (the paper's reference [7, Theorem 3.2]) turns that nondegeneracy into a curve of solutions bending back from $\\kappa^*$, producing the second branch.","core_discovery":"On its own terms, the paper proves that whenever $p>p^*_\\gamma$, $p<p_{JL}$, and $H(p-1)<0$, the solution set of $(L_\\kappa)$ is organized by two constants $0\\le\\kappa_*<\\kappa^*$: the admissible set $K=\\{\\kappa>0: (L_\\kappa)\\text{ has a solution}\\}$ is exactly $(0,\\kappa^*]$, with a minimal solution on $0<\\kappa\\le\\kappa_*$, a unique solution at $\\kappa=\\kappa^*$, no solutions for $\\kappa>\\kappa^*$, and at least two solutions for $\\kappa_*<\\kappa<\\kappa^*$, one of them pointwise above the minimal one. The proof builds a supersolution for small $\\kappa$, shows that minimal solutions are stable, uses new weighted energy estimates to control their decay, passes to the limit to attain $\\kappa^*$, and then applies a bifurcation theorem whose transversality at $\\kappa^*$ forces a backward-folding branch. The boundary data $\\mu$ enter only through growth conditions at the tip and at infinity; the condition $H(p-1)<0$ is exactly the coercivity condition for the weighted estimates once the cone's Hardy constant $((N-2)/2)^2+\\Lambda$ is inserted.","pith_inferences":["The paper leaves the cases $p\\ge p_{JL}$ and $H(p-1)\\ge0$ open. A natural test is to continue the branch numerically past $\\kappa^*$ in one of those regimes: the outcome would show whether the failure of $H(p-1)<0$ corresponds to the endpoint not being attained, the branch not folding, or a different multiplicity pattern; the paper does not predict which.","Because the same cubic $H(p-1)=0$ appears in a related stability/Liouville analysis cited in the paper, the turning point $\\kappa^*$ may coincide with the point where the minimal solution loses linearized stability; the paper does not draw this connection, but its eigenvalue computation at $\\kappa^*$ provides a concrete object to test it.","Since the proof's coercivity depends on the cone's Hardy constant, one would expect the same classification on any domain for which the Hardy constant is $((N-2)/2)^2+\\Lambda$; comparing cones with the same cross-section spectrum but different geometry would show how much of the phenomenon is spectral and how much depends on finer shape."],"forward_implications":["For any cone whose first eigenvalue $\\Lambda$ is known, the full two-threshold picture is valid on the exponent interval $p_1<p<p_{JL}$, where $p_1>p^*_\\gamma$ is the root of $H(p-1)=0$; outside this interval only the existence/nonexistence threshold of Theorem 1.1 is asserted.","At $\\kappa=\\kappa^*$, the unique solution is the limit of the minimal solutions $u_\\kappa$ as $\\kappa\\uparrow\\kappa^*$, so the minimal branch extends continuously to the endpoint.","For $\\kappa_*<\\kappa<\\kappa^*$, the bifurcation branch yields two distinct positive solutions, one strictly larger than the minimal one, so the minimal solution does not exhaust the solution set in that interval.","The same classification holds for Hénon-type equations $-\\Delta u=K(x)u^p$ with $K(x)\\sim |x|^a$, with the cubic replaced by $H_a(p-1)$; in some ranges of $a$ the admissible exponents form a union of two intervals (Theorem 7.2)."],"supporting_citations":[{"why":"Supplies the sharp cone Hardy inequality with constant $((N-2)/2)^2+\\Lambda$, the coercivity input that becomes $H(p-1)<0$.","marker":"[25, Proposition 4.1]"},{"why":"Proves the nonlinear Liouville theorem that sets the lower critical exponent $p^*_\\gamma$ for nonexistence on cones.","marker":"[4]"},{"why":"Provides the supersolution, eigenvalue, and bifurcation strategy adapted here from the supercritical scalar-field equation with forcing.","marker":"[18]"},{"why":"The earlier forcing-term result whose Kelvin-transform decay estimates are replaced by the new cone-adapted weighted estimates.","marker":"[17]"},{"why":"Establishes the half-space version of the two-threshold classification that this paper extends to general cones.","marker":"[20]"},{"why":"The bifurcation theorem used to produce the second solution branch near $\\kappa=\\kappa^*$.","marker":"[7, Theorem 3.2]"},{"why":"The prior appearance of the cubic equation $H(p-1)=0$ in a stability/Liouville analysis of Lane–Emden on cones.","marker":"[8]"}],"fun_headline_variants":["Cone eigenvalue dictates boundary-data thresholds for solutions","Eigenvalue controls existence and multiplicity on cones","Bifurcation at eigenvalue reveals multiple solutions","Existence thresholds tied to cone eigenvalue and boundary data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the cone Hardy inequality with the sharp constant $((N-2)/2)^2+\\Lambda$, cited from [25], is valid on $\\Omega$; this is the exact inequality whose coercivity condition later becomes $H(p-1)<0$, so if the constant were smaller or the inequality failed, the decay estimates behind critical existence and multiplicity would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Cone eigenvalue dictates boundary-data thresholds for solutions","Eigenvalue controls existence and multiplicity on cones","Bifurcation at eigenvalue reveals multiple solutions","Existence thresholds tied to cone eigenvalue and boundary data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000296,"raw_usage":{"total_tokens":1687,"prompt_tokens":882,"completion_tokens":805,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":745}},"tokens_in":498,"tokens_out":805,"duration_ms":7666,"temperature":1.0,"reasoning_tokens":745,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:02:05.720410+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a cone with cross-section $A$ where $H(p-1)<0$ and $p<p_{JL}$, and numerically continue the branch of positive solutions of $(L_\\kappa)$ as $\\kappa$ increases: the theorem predicts a minimal branch up to $\\kappa_*$, a second branch for $\\kappa_*<\\kappa<\\kappa^*$, a single solution at $\\kappa^*$, and none above it. A branch count different from that — for instance, no solution at $\\kappa^*$, three solutions in the interval, or a solution for $\\kappa>\\kappa^*$ — would refute the classification. Separately, checking the best constant in (1.16) for that cone decides whether the external Hardy-inequality input really is sharp.","supporting_citations":[{"cited_title":"Berestycki, I","cited_arxiv_id":null,"evidence_quote":"Proves the nonlinear Liouville theorem that sets the lower critical exponent $p^*_\\gamma$ for nonexistence on cones."},{"cited_title":"Ishige, S","cited_arxiv_id":null,"evidence_quote":"Provides the supersolution, eigenvalue, and bifurcation strategy adapted here from the supercritical scalar-field equation with forcing."},{"cited_title":"Ishige and S","cited_arxiv_id":null,"evidence_quote":"The earlier forcing-term result whose Kelvin-transform decay estimates are replaced by the new cone-adapted weighted estimates."},{"cited_title":"Katayama, Semilinear elliptic problems on the half space with a superc ritical nonlinearity, Discrete Contin","cited_arxiv_id":null,"evidence_quote":"Establishes the half-space version of the two-threshold classification that this paper extends to general cones."},{"cited_title":"Dupaigne, A","cited_arxiv_id":null,"evidence_quote":"The prior appearance of the cubic equation $H(p-1)=0$ in a stability/Liouville analysis of Lane–Emden on cones."}],"review_version":1}