{"id":"ba89d5e8-d717-44e0-9de3-f88aa8b56253","arxiv_id":"2411.11037","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the p-Kirchhoff equation with prescribed L^p mass in R^3, the paper establishes the existence-nonexistence trichotomy, radial ground states and infinitely many high-energy solutions in the supercritical range, and convergence to the p-Laplacian limit as b approaches 0.","lead":"This mathematics paper proves existence, nonexistence, multiplicity, uniqueness, and limit behavior of solutions with a prescribed L^p norm for a p-Kirchhoff equation in three-dimensional space. It gives the full parameter map and explicit mass thresholds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.4's printed subadditivity proof is invalid, and Theorem 1.1's compactness argument depends on it; the lemma is repairable but must be rewritten.","rationale":"The reader correctly spotted Lemma 3.4 as a serious gap and recommended CONDITIONAL; I agree that the gap is local and repairable. I differ slightly in emphasis: the reader's 'weakest assumption' is the external sharp Gagliardo-Nirenberg inequality, whereas I see the invalid subadditivity derivation as the most load-bearing weakness because it is internal and is the exact step that converts a Palais-Smale sequence into a compact minimizer in Theorem 1.1. The GN inequality is cited to Agueh and Weinstein and, together with Serrin-Tang uniqueness for 3/2 < p <= 2, is standard; the proof of Lemma 3.4, as printed, is not. I also noted a radiality slip in the proof of Theorem 1.3, where a minimizing sequence in M(c) subset W^{1,p}_r is translated and then treated as if still radial; this is repairable by invoking radial compactness of W^{1,p}_r into L^q directly, so it does not change my overall conditional assessment. No machine-checked proof or reproducible code is offered, so accepting the paper requires the authors to supply the missing arguments.","tokens_in":25296,"tokens_out":24569,"duration_ms":289160,"concrete_test":"Rewrite the proof of Lemma 3.4 replacing the invalid final display with the monotonicity argument: first prove i(theta c) < theta^p i(c) for theta > 1 by checking the two correction terms are negative exactly when p > 3/2; then set theta = c/alpha and theta = c/beta to get i(c)/c^p < i(alpha)/alpha^p and i(c)/c^p < i(beta)/beta^p, and add the two inequalities. Verify this yields strict subadditivity for every c satisfying (3.8). If the step i(c)/c^p < i(alpha)/alpha^p plus i(c)/c^p < i(beta)/beta^p fails for some threshold case, the contradiction used after Equation (3.19) does not go through.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The existence part of Theorem 1.1 rules out dichotomy by applying Lemma 3.4: for every c satisfying (3.8), i(c) < i(alpha) + i(sqrt[p]{c^p - alpha^p}) for 0 < alpha < c. The proof of Lemma 3.4 begins with the scaling computation (3.9), which is correct for p > 3/2, but the final display rewrites i(c) as a weighted sum of i(c) itself and then asserts strict inequality without a valid argument. The displayed identity is tautological and does not follow from (3.9), so the strict subadditivity that is essential after (3.19) is not established by the printed proof. A repair is available: (3.9) implies the map c |-> i(c)/c^p is strictly decreasing on the range (3.8); then, for alpha^p + beta^p = c^p, applying this monotonicity to alpha and beta and summing gives i(c) = (alpha^p/c^p)i(c) + (beta^p/c^p)i(c) < i(alpha) + i(beta). Since this is the only place where the paper eliminates splitting of a minimizing sequence, Theorem 1.1(1)-(3) is conditional on that repair.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the p-Kirchhoff equation with prescribed L^p mass in R^3, for 3/2<p<3 and p<q<p*. It claims: (i) sharp existence/nonexistence of constrained minimizers of the energy in the L^p-subcritical and L^p-critical ranges, with no minimizer at the L^p-critical exponent; (ii) existence of a radial ground state and of infinitely many radial solutions with diverging energy in the L^p-supercritical range; (iii) convergence of those solutions, as the Kirchhoff coefficient b tends to 0+, to solutions of a p-Laplacian equation; and (iv) uniqueness and explicit formulas for minimizers when 3/2<p<=2. The proofs combine the sharp Gagliardo-Nirenberg inequality of p-Laplacian type, minimization on S(c) and on a Pohozaev manifold, Ekeland's principle, and a Jeanjean-type linking construction.","tokens_in":25341,"tokens_out":25947,"duration_ms":251484,"significance":"If the results hold, the paper gives a fairly complete existence-nonexistence-multiplicity picture for normalized solutions of this p-Kirchhoff problem, extending known p=2 results to the quasilinear setting. The explicit thresholds and minimizers are expressed through the optimizer Q of the Gagliardo-Nirenberg inequality, and the b->0+ asymptotic statement is a useful addition. The variational scheme is standard and the main theorems are plausible. However, several load-bearing proofs contain gaps that need to be repaired: the strict subadditivity in Lemma 3.4 is not proved as written, the displayed critical-mass formula c_* contains an algebraic error, and the radial compactness argument in Theorem 1.3 is not justified. These issues are local and appear repairable, but they affect the central existence and threshold claims.","major_comments":[{"comment":"The proof of the strict subadditivity is invalid as printed. The final display reads i(c) = (alpha^p/c^p) i(c/alpha * alpha) + ((c^p-alpha^p)/c^p) i(beta * c/beta) < i(alpha)+i(beta), but c/alpha*alpha = c and c/beta*beta = c, so the equality is the tautology i(c)=i(c). Applying (3.9) with theta=c/alpha and theta=c/beta gives two inequalities of the form i(c)<(c/alpha)^p i(alpha) and i(c)<(c/beta)^p i(beta); adding them gives 2i(c)<i(alpha)+i(beta), not i(c)<i(alpha)+i(beta). Since (3.19) in the proof of Theorem 1.1 uses exactly this strict subadditivity to rule out dichotomy, the printed argument is incomplete. A repair is available: (3.9) implies that c -> i(c)/c^p is strictly decreasing on the range satisfying i(c)<0; applying this monotonicity to alpha and beta and adding the weighted inequalities yields i(c)<i(alpha)+i(beta). The lemma should be rewritten accordingly.","section":"Lemma 3.4"},{"comment":"The displayed formula for c_* contains an algebraic error. With p_2 = 3(q-p)/p^2 - 1, one has 2p p_2 = (6q - 6p - 2p^2)/p, so the factor coming from the Young inequality is (bp/(6q - 6p - 2p^2))^{p_2}, not (bp/(6pq - 8p^2))^{p_2}. For p=2, q=4 these two expressions differ by a factor of 4. Since c_* is the sharp threshold in Theorem 1.1(iii) and in Lemma 3.1, this is not a harmless typo: as printed, the threshold formula is wrong. Please correct the denominator in Theorem 1.1, Lemma 3.1, and anywhere else the formula appears.","section":"Lemma 3.1(4) and Theorem 1.1(iii)"},{"comment":"After finding y_n such that the integral over B_1(y_n) is positive, the proof sets u_n(x)=v_n(x+y_n) and states that {u_n} is a minimizing sequence for m(c). This is false when y_n is nonzero: translations of radial functions are not radial, while M(c) is contained in S_r(c). The subsequent weak limit is claimed in W^{1,p}_r(R^3), which requires the sequence to be radial. If the y_n are unbounded, the translated sequence may even converge weakly to zero. The proof should instead use the Strauss radial compactness theorem to obtain strong L^q convergence directly from the radial minimizing sequence, or otherwise justify the use of translations without destroying radial symmetry.","section":"Proof of Theorem 1.3"},{"comment":"The inference 'Since |u_k|_q^q -> |u|_q^q != 0, we obtain lambda < 0' is not justified. Equation (5.11) gives only lambda <= 0. If lambda = 0, then the limiting equation (5.14) combined with the Nehari identity (5.16) and the Pohozaev identity would force q = p*, contradicting the assumption q < p*. This missing argument is needed both for the negativity of the Lagrange multiplier and for the strong convergence step (5.17). Please add the argument or cite the analogous reasoning used later in Theorem 1.5.","section":"Lemma 5.8(ii)"}],"minor_comments":[{"comment":"There are several typographical errors, including 'Gargliardo' in the abstract, 'rencent' in Section 1, 'exsists' in the proof of Theorem 1.1, and 'Fisrtly' in Section 5; these should be corrected.","section":"Throughout"},{"comment":"The displayed formula for c_* has an unmatched parenthesis after 2p^2 - 3q + 3p; please typeset the formula with consistent parentheses and exponents.","section":"Theorem 1.1 and Lemma 3.1"},{"comment":"In the proof of Theorem 1.6, the coefficient alpha is printed as c/|Q|_P; the subscript should be p.","section":"Theorem 1.6"},{"comment":"The sentence defining t_0 after the Young inequality is garbled in the typesetting; please rewrite it so that the equality condition in (3.7) is stated clearly.","section":"Lemma 3.1(4)"},{"comment":"The notation for the p-th root appears as 'p√' in the statement of Lemma 3.4; use a proper root symbol such as \\sqrt[p]{\\cdot}.","section":"Lemma 3.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely accept-worthy after a major revision. The central variational strategy is standard, but the proof of Lemma 3.4 is invalid as written, the c_* formula contains a concrete algebraic error, and the radial compactness argument in Theorem 1.3 needs to be repaired. These are load-bearing but local issues; I see no reason to doubt the novelty or the overall plausibility of the results. The authors should also double-check the few other places where scaling and radial symmetry interact."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2411.11037. First, it is genuinely the first paper to handle the p-Kirchhoff equation with prescribed L^p mass across the full subcritical-critical-supercritical range, and it gives a fairly complete picture: sharp mass thresholds for minimizers, nonexistence at the L^p-critical exponent, radial ground states and a high-energy sequence in the supercritical case, and a b→0+ limit to the p-Laplacian. Second, the proof of Lemma 3.4, which rules out dichotomy and is load-bearing for Theorem 1.1, is printed incorrectly: the scaling identity in (3.9) does not produce the claimed inequality, and the final display is tautological. The lemma is still true and easily repairable — applying the dilation v = θu gives i(θc) < θ^p i(c) when i(c) < 0, hence c ↦ i(c)/c^p is strictly decreasing, and subadditivity follows. So the paper's main argument survives, but the written proof needs rewriting.\n\nWhat is good: the variational scheme is standard and executed with care. The use of the sharp GN inequality with optimizer Q is honest; the thresholds, explicit minimizers, and mountain-pass levels are all expressed through Q, not through quantities of the target equation, so there is no circularity. The uniqueness statements for 3/2 < p ≤ 2 properly rely on the known uniqueness of Q, and the paper acknowledges the p > 2 obstruction. The b→0+ asymptotic section is a nice add-on, and the multiplicity argument follows the Jeanjean-Bartsch linking framework cleanly.\n\nSoft spots: aside from Lemma 3.4, there is a typo in the proof of Theorem 1.1(3) where a sequence is claimed to lie in S(c) but should be in S(c_n). Several auxilary lemmas are omitted by reference; that is acceptable if the cited analogues really match, but the authors should spell out the adaptation for the nonlocal term. The dependence on the sharp GN inequality is an external input, and if equality cases for p > 2 were not unique, the explicit formulas in Theorem 1.6 would not extend; the paper is clear about that.\n\nBottom line: this is a serious paper for the normalized-solutions community. It deserves a real referee and will probably be cited once the proof of Lemma 3.4 is corrected. I would send it to peer review, but require the authors to replace that proof and fix the small errors before it is accepted.","headline":"Genuinely first full normalized-solution picture for the p-Kirchhoff equation in R^3, but the paper ships a wrong proof of a load-bearing lemma that is nevertheless repairable.","tokens_in":26109,"tokens_out":7868,"would_cite":true,"duration_ms":72825,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J20","35J92","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The p-Kirchhoff equation with prescribed mass has a complete existence-nonexistence-multiplicity picture, governed by two critical exponents and a sharp Gagliardo-Nirenberg optimizer.","keywords":["p-Kirchhoff equation","normalized solutions","prescribed L^p mass","ground state","multiplicity","asymptotic behavior","Gagliardo-Nirenberg inequality","p-Laplacian"],"falsifier":"Numerically minimize the constrained energy for one fixed $p\\in(3/2,2)$ and $q=p+2p^2/3$ across a range of masses $c$: Theorem 1.1(2) predicts the infimum is $0$ for $c$ below the stated threshold and $i(c)=-\\infty$ above, with no minimizer in either case. Finding a finite attained minimum for any $c>0$ would falsify the central nonexistence claim.","tokens_in":24887,"feed_emoji":"📐","tokens_out":26687,"duration_ms":219068,"temperature":0.7,"pith_summary":"This paper studies the p-Kirchhoff equation $-\\left(a+b\\int_{\\mathbb{R}^3}|\\nabla u|^p dx\\right)\\Delta_p u=\\lambda|u|^{p-2}u+|u|^{q-2}u$ in $\\mathbb{R}^3$ with prescribed $L^p$ norm $\\|u\\|_p=c$, where $3/2<p<3$ and $p<q<p^*$. Its aim is to decide, for every mass $c>0$, whether constrained critical points exist, how many there are, and what happens as the nonlocal coefficient $b$ tends to $0$. The answer depends on where $q$ sits relative to the thresholds $p+p^2/3$ and $p+2p^2/3$: minimizers of the energy exist exactly above stated masses in the subcritical and intermediate regimes, no minimizer exists at $q=p+2p^2/3$, and in the supercritical range there is a radial ground state for every $c>0$ together with infinitely many radial solutions whose energies tend to $+\\infty$. The sharp formulas for the thresholds and explicit minimizers are built on a Gagliardo-Nirenberg inequality of p-Laplacian type and its optimizer $Q$; when $3/2<p\\le2$, uniqueness of $Q$ up to translations gives uniqueness of the minimizers and an exact mountain-pass level.","feed_headline":"Thresholds decide every normalized p-Kirchhoff solution","feed_subtitle":"One sharp inequality fixes when solutions exist, how many there are, and their limit as the nonlocal term vanishes.","key_machinery":"The load-bearing object is the sharp Gagliardo-Nirenberg inequality of p-Laplacian type (Lemma 2.1), which bounds $\\|u\\|_q$ by a power of $\\|\\nabla u\\|_p$ times a power of $\\|u\\|_p$, with the sharp constant expressed through the ground state $Q$ of the auxiliary equation (2.2). This inequality supplies the lower bounds that reduce the constrained energy to a one-variable function of $t=\\|\\nabla u\\|_p^p$; all threshold masses, including $a^{3/p^2}\\|Q\\|_p$, $c_*$, and the critical-mass condition in Lemma 3.2, are read off from that reduced function. The other central mechanism is the Pohozaev identity $P(u)=a\\|\\nabla u\\|_p^p+b\\|\\nabla u\\|_p^{2p}-\\frac{3(q-p)}{pq}\\|u\\|_q^q=0$, the integral relation any weak solution must satisfy; it defines the natural constraint manifold $\\mathcal{M}(c)$ used in the supercritical case and forces $\\lambda<0$ for every solution. In the multiplicity proof, a scaling map $k(u,\\theta)=e^{3\\theta/p}u(e^{\\theta}x)$ and a finite-dimensional linking argument produce Palais-Smale sequences with $P(u_k)\\to0$; convergence of those sequences is what yields the high-energy radial solutions.","core_discovery":"The central discovery is that the constrained minimization problem for the energy $I(u)=\\frac{a}{p}\\int_{\\mathbb{R}^3}|\\nabla u|^p\\,dx+\\frac{b}{2p}\\left(\\int_{\\mathbb{R}^3}|\\nabla u|^p\\,dx\\right)^2-\\frac{1}{q}\\int_{\\mathbb{R}^3}|u|^q\\,dx$ on the sphere $\\|u\\|_p=c$ is governed by a sharp dichotomy in $q$. For $p<q<p+p^2/3$, $i(c)$ has a minimizer for every $c>0$; for $q=p+p^2/3$, it has a minimizer exactly for $c>a^{3/p^2}\\|Q\\|_p$; for $p+p^2/3<q<p+2p^2/3$, there is a critical mass $c_*>0$ such that minimizers exist exactly for $c\\ge c_*$; and for $q=p+2p^2/3$, no minimizer exists for any $c>0$. In the supercritical range $p+2p^2/3<q<p^*$, where $I$ is unbounded below on $\\mathcal{S}(c)$, the paper works on the radial space and obtains, for every $c>0$, a radial ground state and a sequence of radial solutions with $\\|u_n\\|_{W^{1,p}_r}\\to+\\infty$ and $I(u_n)\\to+\\infty$; as $b\\to0^+$, these solutions converge in $W^{1,p}_r(\\mathbb{R}^3)$ to weak solutions of the limiting p-Laplacian equation $-a\\Delta_p u-\\lambda|u|^{p-2}u=|u|^{q-2}u$. For $3/2<p\\le2$, the unique minimizer is explicit: $u_c=c\\,\\mu_q^{3/p}\\|Q\\|_p^{-1}Q(\\mu_q x)$ with $\\mu_q$ determined by a one-variable function $f_q$, and the mountain-pass value $\\gamma(c)=f_q(t_q)$ is attained by the same scaling.","pith_inferences":["An implication the paper leaves implicit is that, for $3/2<p\\le2$, the explicit minimizer formula forces every subcritical and minimizer solution to be a scaling of the single optimizer $Q$; uniqueness there is structural rather than accidental.","A testable extension is to measure the rate of convergence in $b\\to0^+$ for the first few radial solutions; the proof gives $W^{1,p}_r$ convergence but no rate, and the rate may depend on the energy level.","The critical exponent $q=p+2p^2/3$ is exactly where the scaling exponent $3(q-p)/p$ in the Pohozaev term crosses the exponent $2p$ of the nonlocal term; the same crossing should produce analogous existence/nonexistence dichotomies in other nonlocal problems.","For $p>2$, the missing uniqueness of the optimizer $Q$ blocks the explicit formulas; if that uniqueness were established, the same argument would extend Theorems 1.6 and 1.8 beyond $p\\le2$."],"forward_implications":["For $p<q<p+p^2/3$, every prescribed mass $c>0$ admits a minimizer, so the nonlocal Kirchhoff term does not destroy the subcritical existence picture.","At $q=p+p^2/3$ and in $p+p^2/3<q<p+2p^2/3$, minimizers exist only above explicit mass thresholds, namely $c>a^{3/p^2}\\|Q\\|_p$ and $c\\ge c_*$ respectively.","At $q=p+2p^2/3$, the infimum is never attained for any $c>0$; any normalized solution at this critical exponent must come from a different variational principle such as the mountain-pass level of Theorem 1.8.","In the supercritical range $p+2p^2/3<q<p^*$, radial ground states exist for every $c>0$, and there are infinitely many radial solutions with energies tending to $+\\infty$.","As $b\\to0^+$, each of these radial solutions converges (up to subsequence) to a weak solution of the limiting p-Laplacian equation with the same prescribed mass."],"supporting_citations":[{"why":"This reference supplies the sharp Gagliardo-Nirenberg inequality of p-Laplacian type used to bound the energy and compute all thresholds.","marker":"[10]"},{"why":"This reference identifies the ground state Q of the auxiliary p-Laplacian equation that realizes equality in the inequality.","marker":"[11]"},{"why":"This reference gives the uniqueness up to translations of the ground state Q for 1<p≤2, on which Theorems 1.6 and 1.8 rely.","marker":"[24]"},{"why":"This reference provides the Pohozaev identity and the constrained-minimization framework that the subcritical and critical proofs follow.","marker":"[12]"},{"why":"This reference supplies Ekeland's variational principle and the vanishing lemma used to turn minimizing sequences into Palais-Smale sequences and prevent loss of mass.","marker":"[14]"},{"why":"This reference supplies the Lagrange multiplier rule used to identify the limit in the critical-mass case of Theorem 1.1.","marker":"[17]"},{"why":"This reference supplies the finite-dimensional linking/min-max scheme and the Palais-Smale sequence with P(u_k)→0 used in the multiplicity proof.","marker":"[19]"},{"why":"This reference gives the linking lemma guaranteeing that every path in Γ_n meets the sphere B_n, producing the lower bound β_n.","marker":"[21]"},{"why":"This reference supplies the auxiliary functional I∘k on S(c)×R that turns the constrained problem into an unconstrained min-max.","marker":"[22]"},{"why":"This reference supplies the weak-convergence fact used to pass to the limit in the asymptotic and multiplicity arguments.","marker":"[23]"}],"fun_headline_variants":["Sharp q-thresholds dictate p-Kirchhoff normalized solutions","Criticality determines existence, count, and vanishing nonlocal limit","p-Kirchhoff: one exponent rules all normalized solutions","Asymptotic limits and multiplicity from p-Kirchhoff thresholds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's thresholds, explicit minimizers, and uniqueness claims all rest on a sharp interpolation inequality whose optimizer is the ground state $Q$; for $3/2<p\\le2$, it also relies on $Q$ being unique up to translation, and these external facts are cited rather than reproved.","fun_headline_variants_meta":{"raw":{"variants":["Sharp q-thresholds dictate p-Kirchhoff normalized solutions","Criticality determines existence, count, and vanishing nonlocal limit","p-Kirchhoff: one exponent rules all normalized solutions","Asymptotic limits and multiplicity from p-Kirchhoff thresholds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000678,"raw_usage":{"total_tokens":3294,"prompt_tokens":1371,"completion_tokens":1923,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":987,"completion_tokens_details":{"reasoning_tokens":1862}},"tokens_in":987,"tokens_out":1923,"duration_ms":16817,"temperature":1.0,"reasoning_tokens":1862,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:01:15.916731+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically minimize the constrained energy for one fixed $p\\in(3/2,2)$ and $q=p+2p^2/3$ across a range of masses $c$: Theorem 1.1(2) predicts the infimum is $0$ for $c$ below the stated threshold and $i(c)=-\\infty$ above, with no minimizer in either case. Finding a finite attained minimum for any $c>0$ would falsify the central nonexistence claim.","supporting_citations":[{"cited_title":"Sharp gagliardo–nirenberg inequalities via p-laplacian type equations,","cited_arxiv_id":null,"evidence_quote":"This reference supplies the sharp Gagliardo-Nirenberg inequality of p-Laplacian type used to bound the energy and compute all thresholds."},{"cited_title":"Nonlinear schr¨ odinger equations and sharp interpolation estimates,","cited_arxiv_id":null,"evidence_quote":"This reference identifies the ground state Q of the auxiliary p-Laplacian equation that realizes equality in the inequality."},{"cited_title":"Uniqueness of ground states for quasilinear elliptic equations,","cited_arxiv_id":null,"evidence_quote":"This reference gives the uniqueness up to translations of the ground state Q for 1<p≤2, on which Theorems 1.6 and 1.8 rely."},{"cited_title":"Willem, Minimax theorems, vol","cited_arxiv_id":null,"evidence_quote":"This reference supplies Ekeland's variational principle and the vanishing lemma used to turn minimizing sequences into Palais-Smale sequences and prevent loss of mass."},{"cited_title":"Chang, Methods in nonlinear analysis , vol","cited_arxiv_id":null,"evidence_quote":"This reference supplies the Lagrange multiplier rule used to identify the limit in the critical-mass case of Theorem 1.1."},{"cited_title":"Normalized solutions to p-laplacian equations with combined nonlinear- ities,","cited_arxiv_id":null,"evidence_quote":"This reference supplies the finite-dimensional linking/min-max scheme and the Palais-Smale sequence with P(u_k)→0 used in the multiplicity proof."},{"cited_title":"Normalized solutions of nonlinear Schr\\\"odinger equations","cited_arxiv_id":"1209.0950","evidence_quote":"This reference gives the linking lemma guaranteeing that every path in Γ_n meets the sphere B_n, producing the lower bound β_n."},{"cited_title":"Existence of solutions with prescribed norm for semilinear elliptic equations,","cited_arxiv_id":null,"evidence_quote":"This reference supplies the auxiliary functional I∘k on S(c)×R that turns the constrained problem into an unconstrained min-max."},{"cited_title":"Willem, Functional analysis: Fundamentals and applications","cited_arxiv_id":null,"evidence_quote":"This reference supplies the weak-convergence fact used to pass to the limit in the asymptotic and multiplicity arguments."}],"review_version":1}