{"id":"9c5c99b6-341d-4a11-b905-7bbd8d0076a7","arxiv_id":"2411.10317","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The masses of action and nodal action ground states form a full interval, which yields normalized nodal solutions and least-energy/action-ground-state identifications on bounded domains.","lead":"This paper develops a new variational method for nonlinear Schrödinger equations with prescribed mass on bounded domains, and proves that the set of masses admitted by action ground states is always an interval. This interval description yields new existence results for sign-changing solutions and identifies when least energy normalized solutions coincide with action ground states.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Nodal mass characterization is proved only for connected domains; Theorem 1.1's stated generality to arbitrary bounded open sets is unsupported because Proposition 2.8's connected exhaustion is impossible for disconnected Ω.","rationale":"I re-checked the central derivative characterization in the connected case, where the intended problem is set. Proposition 3.2's identity (3.3) follows from the same perturbation argument used for the signed case, and Lemma 4.2's minimizer argument is correct: the inequalities from the interior minimum force both one-sided difference quotients to converge to μ/2, so the derivative exists and equals μ. Lemma 4.6 and the supercritical endpoint argument are coherent. The only genuine gap I found is the disconnected-domain overclaim: Proposition 2.8's exhaustion by connected smooth sets is impossible for disconnected Ω, and every subsequent compactness statement (Proposition 2.7) is stated for connected Ω_n. Since the paper's standing assumption at the start of Section 1 is that Ω is connected, this does not threaten the main results for the intended domains, but it does mean Theorem 1.1 as phrased is not proved. This matches the reader's CONDITIONAL verdict, so I do not adjust it.","tokens_in":21568,"tokens_out":40929,"duration_ms":388819,"concrete_test":"Take Ω = B_1 ∪ B_2 with two disjoint balls. Derive J_nod_Ω(λ) and M_nod_p(Ω) by componentwise reduction: prove J_nod_Ω(λ) = 2 min_i J_{Ω_i}(λ) and M_nod_p(Ω) = M_p(Ω_1) + M_p(Ω_2). If this reduction holds, the interval conclusion survives for disconnected domains and the gap is presentational; if it fails, Theorem 1.1 is false in its stated 'open and bounded' generality. A simpler check is to rerun Proposition 2.8's exhaustion argument with Ω_n taken to be the disjoint union of smooth connected approximations of each component and verify Proposition 2.7's conclusion still passes to the union.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 is stated for every bounded open Ω, but the proof of the nodal case silently assumes connectedness. Proposition 2.8 asserts that an arbitrary bounded open set can be exhausted by connected smooth domains Ω_n with Ω_n⊂Ω_{n+1} and ∪Ω_n=Ω; this is impossible when Ω is disconnected, since a connected subset of Ω cannot meet two different components. Proposition 2.7, used in Lemma 3.1, Proposition 3.2, and the endpoint argument in Theorem 1.1, is explicitly stated for connected Ω_n. Hence the compactness of nodal ground states and the derivative bracketing (3.3), on which Lemma 4.2 rests, are not proved in the stated generality. For connected Ω the argument is internally consistent: the interior-minimizer argument in Lemma 4.2 correctly combines the one-sided bounds from (3.3) to force differentiability with derivative μ/2, and the supercritical endpoint is handled by Proposition 2.7. The overclaim does not undermine the intended connected-domain results, but Theorem 1.1 as written is not fully established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an action-based approach to normalized solutions of the nonlinear Schr\\\"odinger equation with Dirichlet boundary conditions on bounded domains. For the action functional with parameter \\lambda, it studies the masses of positive and nodal action ground states, i.e. the sets M_p(\\Omega) and M_p^{nod}(\\Omega) defined in (1.3). The main result, Theorem 1.1, characterizes these sets as (0,\\infty) in the L^2-subcritical case, as (0,\\mu_p) or (0,\\mu_p] (and analogously for the nodal threshold) in the L^2-critical case, and as (0,\\mu_p] (and (0,\\mu_p^{nod}]) in the L^2-supercritical case. The proof is based on a Darboux-type property for the derivative of the action ground-state level, obtained from one-sided derivative inequalities (3.3) and an interior-minimizer argument. The paper then derives existence of normalized nodal solutions for all masses in the subcritical case and for an interval of masses in the critical and supercritical cases (Theorem 1.2), identifies least energy normalized nodal solutions as nodal action ground states in subcritical and low-mass critical regimes (Theorem 1.4), and proves analogous results on smooth star-shaped domains in the supercritical case (Theorem 1.7). The arguments are detailed and, for connected domains, internally coherent. However, the stated generality over every bounded open set is not supported by the nodal proofs, which rely on connectedness in several places.","tokens_in":21773,"tokens_out":17122,"duration_ms":175958,"significance":"If the characterization is correct in its stated form, it is a substantial contribution: it gives the first complete description of the mass range of action ground states and nodal action ground states, yields new existence results for normalized nodal solutions, and clarifies when least energy normalized solutions coincide with action ground states. The approach is original and potentially adaptable to other problems admitting a Nehari-manifold structure. The paper contains detailed proofs, and the central interval characterization is a genuine variational statement rather than a tautology. A notable strength is that the argument does not require a C^1 branch of solutions and instead uses only one-sided derivative bounds, which is a significant methodological improvement. The main caveat is the disconnected-domain gap: the nodal part of the proof requires connectedness through Proposition 2.7 and Proposition 2.8, so the theorems as stated are not fully established.","major_comments":[{"comment":"The nodal case is proved only under a connectedness assumption, whereas Theorems 1.1, 1.2 and 1.4 are stated for every bounded open set. Proposition 2.8 asserts that an arbitrary bounded open set \\Omega can be exhausted by connected smooth open sets \\Omega_n with \\Omega_n \\subset \\Omega_{n+1} and \\cup_n \\Omega_n = \\Omega, citing [10, Proposition 8.2.1]. This is impossible when \\Omega is disconnected, because a connected subset of \\Omega is contained in a single connected component. Consequently Proposition 2.7, which explicitly assumes connected \\Omega_n, cannot supply the compactness and convergence results on which Lemma 3.1, Proposition 3.2 and the endpoint argument in the proof of Theorem 1.1 rely. The derivative bracketing (3.3) and the existence of nodal action ground states are therefore not established for disconnected domains. Since the problem statement in the introduction restricts \\Omega to be connected, the theorems should either be restricted to connected bounded open sets or supplied with a separate component-wise argument.","section":"Theorem 1.1 and Proposition 2.8"},{"comment":"The proof of Proposition 2.4 uses a sign-changing second eigenfunction \\phi_2 and the identity \\lambda_2(\\Omega) = \\lambda_1(\\text{supp }\\phi_2^+) = \\lambda_1(\\text{supp }\\phi_2^-). For a disconnected domain, \\lambda_2(\\Omega) can be the first eigenvalue of one component, and a corresponding eigenfunction may be one-signed, so this identity is not valid in the generality claimed in Theorem 1.1. This is a second concrete manifestation, independent of the exhaustion argument, of the fact that the nodal theory in Section 2 is developed for connected domains.","section":"Proposition 2.4 and disconnected domains"}],"minor_comments":[{"comment":"The proof uses a density statement asserting that a nodal action ground state can be approximated by functions in N_\\lambda^{nod}(\\Omega) \\cap C_c^\\infty(\\Omega) with arbitrarily close action. This is plausible but not immediate; a short justification via separate approximation of the positive and negative parts with disjoint supports would make the argument self-contained.","section":"Proposition 2.7"},{"comment":"The proof uses Lemma 4.2 for all \\mu \\leq \\tilde{\\mu}_p^{nod}, but the stated inequality \\tilde{\\mu}_p^{nod} \\leq \\mu_p^{nod} alone does not exclude equality. Since the action level has power-type behavior near -\\lambda_2, the inequality is in fact strict, and it would be helpful to state and prove this strictness so that the appeal to Lemma 4.2 is fully transparent.","section":"Theorem 1.7, Step 2"},{"comment":"In the statements of Theorem 1.1 and in the displayed definitions (1.3), the sets M_p(\\Omega) and M_p^{nod}(\\Omega) are sometimes written without the domain \\Omega, and the tilded constants in Theorem 1.7 and its proof are typeset inconsistently. A uniform notation would improve readability.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the mismatch between the stated domain class and the connectedness assumptions actually used in the nodal proofs. If the authors confirm that the intended setting is connected bounded open sets and revise the theorem statements accordingly, the central results appear sound and the paper would be a strong contribution. The reliance on [11], which involves two of the authors, is a standard use of a published result and does not constitute circularity. I would encourage the authors to state the exact class of domains in every theorem and to make the strictness argument in Theorem 1.7 explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core is a new and useful idea: instead of tracking the mass of a chosen curve of ground states, the authors characterize the set of masses as the range of the derivative of the action level J_Ω (resp. J_nod_Ω), using one-sided derivative bounds and a minimization argument. That gives a Darboux-type theorem and, as a payoff, existence of normalized nodal solutions for all masses in the subcritical case and an interval in the critical and supercritical cases, plus identification of least energy normalized solutions with action ground states in the right regimes. The proofs are careful, and the dependence on [11] is legitimate; the imported asymptotic and derivative facts are published and exactly the right tools.\n\nThe soft spot is the stated generality. Theorem 1.1 is announced for every bounded open Ω, but the nodal case is proved only for connected domains. Proposition 2.8 claims an exhaustion by connected smooth domains Ω_n for an arbitrary bounded open set; that is impossible if Ω is disconnected, since a connected subset cannot meet two components. Proposition 2.7, which is used to get compactness and the derivative bracketing (3.3), explicitly assumes connected Ω_n. So the proof does not cover disconnected domains. This is not a fatal flaw for the problem as it is usually stated—the introduction restricts to connected bounded open sets—but the theorem statement overreaches, and the gap is real. It can likely be fixed componentwise, but as written the generality is unsupported. A referee should ask for that fix.\n\nThe rest holds up: for connected domains, the chain from one-sided derivative inequalities to the interval characterization is coherent, the endpoint argument in the supercritical case works, and the star-shaped result in Theorem 1.7 is a genuine advance over the ball-specific result in [18]. The distinction between (0, μ) and (0, μ] in the critical case is left open, which is honest rather than a flaw.\n\nI would send this to peer review. It deserves referee time: the method is transferable, the results answer real open questions, and the flaw is a boundary condition on the statement, not a crack in the central argument.","headline":"A genuinely new Darboux-type mass characterization for normalized NLS solutions, with a real overclaim on disconnected domains that does not undermine the connected-domain results.","tokens_in":22298,"tokens_out":2934,"would_cite":true,"duration_ms":29752,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","49J40","58E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the set of masses of action ground states, and of nodal action ground states, is always a full interval on bounded domains, and uses this to produce normalized solutions of prescribed mass.","keywords":["nonlinear Schrödinger equations","normalized solutions","nodal solutions","action ground states","Nehari manifold","Darboux property","least energy normalized solutions","bounded domains"],"falsifier":"Take a specific bounded domain, for example an interval in $N=1$ with $p>6$, and compute the masses of nodal action ground states for all $\\lambda>-\\lambda_2$. Theorem 1.1(iii) predicts the set of masses is exactly $(0,\\mu_p^{\\rm nod}]$ with no gaps and the endpoint attained. Finding a mass $\\mu<\\mu_p^{\\rm nod}$ with no nodal solution, or a gap in the set of masses, would falsify the characterization.","tokens_in":21379,"feed_emoji":"🧮","tokens_out":7530,"duration_ms":65846,"temperature":0.7,"pith_summary":"This paper proves a complete interval characterization for the masses of action ground states and of nodal action ground states of the nonlinear Schrödinger equation on any bounded open domain. For the subcritical exponent $p<2+4/N$ every positive mass occurs; for the critical exponent $p=2+4/N$ the masses fill $(0,\\mu_p)$ or $(0,\\mu_p]$ (and similarly for the nodal set); for the supercritical case they fill $(0,\\mu_p]$ and $(0,\\mu_p^{\\rm nod}]$. The characterization is obtained not by following a smooth curve of solutions but by showing that the mass set is the range of the derivative of the ground-state level, a Darboux-type property. This gives normalized nodal solutions for every mass in the subcritical regime and for a whole interval of masses in the critical and supercritical regimes, and identifies least-energy normalized (nodal) solutions as action (nodal action) ground states in those regimes. The approach is designed to extend to other Nehari-manifold problems.","feed_headline":"Mass sets of NLS ground states are exactly intervals","feed_subtitle":"A frequency-level derivative gives sharp mass thresholds, yielding nodal solutions for every mass in the subcritical range.","key_machinery":"The central object is the action functional $J_\\lambda(u,\\Omega)=\\frac12\\|\\nabla u\\|_2^2+\\frac{\\lambda}{2}\\|u\\|_2^2-\\frac1p\\|u\\|_p^p$ on $H^1_0(\\Omega)$, restricted to the Nehari manifold $\\mathcal{N}_\\lambda(\\Omega)$ or its nodal analogue $\\mathcal{N}_\\lambda^{\\rm nod}(\\Omega)$. The level functions $J_\\Omega(\\lambda)$ and $J_\\Omega^{\\rm nod}(\\lambda)$ are locally Lipschitz and strictly increasing on their natural frequency ranges, and the key inequality (3.3) says that the one-sided derivatives of $J_\\Omega^{\\rm nod}$ bracket half the masses of its ground states. For a prescribed mass $\\mu$, the paper minimizes $f_\\mu(\\lambda)=J_\\Omega^{\\rm nod}(\\lambda)-\\frac{\\mu}{2}\\lambda$; any interior minimizer is a point where $J_\\Omega^{\\rm nod}$ is differentiable with derivative $\\mu$, so the corresponding ground state has exactly the prescribed mass. This converts an existence problem into a one-dimensional calculus problem.","core_discovery":"On the paper's own terms, Theorem 1.1 states that for every bounded open $\\Omega\\subset\\mathbb{R}^N$ and every $p\\in(2,2^*)$, with $2^*=2N/(N-2)$, the sets $M_p(\\Omega)$ and $M_p^{\\rm nod}(\\Omega)$ of $L^2$-masses of action ground states and nodal action ground states are exactly the following: $(0,\\infty)$ when $p<2+4/N$; $(0,\\mu_p)$ or $(0,\\mu_p]$ (and the analogous alternative for the nodal set) when $p=2+4/N$; and $(0,\\mu_p]$ and $(0,\\mu_p^{\\rm nod}]$ when $p>2+4/N$. The thresholds $\\mu_p$ and $\\mu_p^{\\rm nod}$ are finite in the critical and supercritical regimes and are realized as suprema of the derivative of the corresponding ground-state level. From this, Theorem 1.2 gives normalized nodal solutions for every mass in the subcritical case and for every mass up to the threshold in the critical and supercritical cases. Theorem 1.4 shows that under the same mass ranges least-energy normalized nodal solutions are exactly nodal action ground states; Theorem 1.7 does the analogous identification for small masses on smooth star-shaped domains in the supercritical case, with frequencies uniformly bounded above.","pith_inferences":["Beyond the paper, the same level-derivative mechanism should identify mass thresholds for other equations with a Nehari manifold, for example problems with combined nonlinearities or Neumann boundary conditions, because the proof never uses a smooth curve of solutions.","The critical-regime gap between $2\\mu_N$ and $\\mu_p^{\\rm nod}$ is an explicit open target: deciding whether least-energy normalized nodal solutions with masses in that range remain nodal action ground states would complete Theorem 1.4.","For non-star-shaped supercritical domains, Theorem 1.7 is open; an annulus or another non-star-shaped domain is the natural place to test whether the small-mass identification and frequency bound survive."],"forward_implications":["For every bounded open domain and every $p<2+4/N$, normalized nodal solutions exist for every prescribed mass $\\mu>0$ (Theorem 1.2(i)).","For $p=2+4/N$, normalized nodal solutions exist for every $\\mu\\in(0,\\mu_p^{\\rm nod})$; for $p>2+4/N$ they exist for every $\\mu\\in(0,\\mu_p^{\\rm nod}]$, where $\\mu_p^{\\rm nod}$ is the finite threshold of Theorem 1.1 (Theorem 1.2(ii)-(iii)).","In the subcritical regime (all $\\mu>0$) and in the critical regime with $\\mu<2\\mu_N$, every least-energy normalized nodal solution is a nodal action ground state for the frequency it carries (Theorem 1.4).","On smooth star-shaped domains with $p>2+4/N$, least-energy normalized (nodal) solutions exist for every mass up to the respective threshold, and for sufficiently small masses they are action (nodal action) ground states with frequencies bounded by a constant depending only on $p$ and $\\Omega$ (Theorem 1.7).","When $\\Omega$ is a ball and $p=2+4/N$, masses in $[\\mu_N,2\\mu_N)$ admit least-energy normalized nodal solutions while no positive normalized solution exists, so the least-energy solutions are necessarily nodal (Remark 1.6)."],"supporting_citations":[{"why":"Supplies the action-ground-state level properties (existence, monotonicity, one-sided derivative bounds) that the mass characterization extends to the nodal case.","marker":"[11]"},{"why":"Gives existence of nodal action ground states on smooth domains, which the paper extends to arbitrary bounded open sets.","marker":"[4]"},{"why":"Provides the uniform lower bound argument for the nodal level used in Proposition 2.5 and in the compactness estimates.","marker":"[7]"},{"why":"Supplies the exhaustion of an arbitrary bounded open set by smooth bounded open sets, used with Proposition 2.7 to transfer nodal ground states to rough domains.","marker":"[10]"},{"why":"Establishes that nodal action ground states are actual solutions of the equation, the bridge from Nehari sets to the PDE.","marker":"[5]"},{"why":"Gives the ball result on normalized supercritical solutions used as comparison and in Remark 1.6 for critical least-energy nodal solutions.","marker":"[18]"},{"why":"Supplies the Pohozaev identity used on star-shaped domains to bound the energy of solutions from below in Theorem 1.7.","marker":"[26]"}],"fun_headline_variants":["Exact mass ranges for NLS action states","Every mass works for subcritical NLS nodal solutions","NLS ground-state masses: full characterization","Thresholds that decide NLS normalized solutions","When nodal and ground states coincide in NLS"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the ground-state level functions, especially the nodal one, are locally Lipschitz and have one-sided derivatives at every frequency that bracket the masses of the corresponding ground states, as in inequality (3.3).","fun_headline_variants_meta":{"raw":{"variants":["Exact mass ranges for NLS action states","Every mass works for subcritical NLS nodal solutions","NLS ground-state masses: full characterization","Thresholds that decide NLS normalized solutions","When nodal and ground states coincide in NLS"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000336,"raw_usage":{"total_tokens":1867,"prompt_tokens":955,"completion_tokens":912,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":842}},"tokens_in":571,"tokens_out":912,"duration_ms":9087,"temperature":1.0,"reasoning_tokens":842,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:46:28.745111+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a specific bounded domain, for example an interval in $N=1$ with $p>6$, and compute the masses of nodal action ground states for all $\\lambda>-\\lambda_2$. Theorem 1.1(iii) predicts the set of masses is exactly $(0,\\mu_p^{\\rm nod}]$ with no gaps and the endpoint attained. Finding a mass $\\mu<\\mu_p^{\\rm nod}$ with no nodal solution, or a gap in the set of masses, would falsify the characterization.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the action-ground-state level properties (existence, monotonicity, one-sided derivative bounds) that the mass characterization extends to the nodal case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives existence of nodal action ground states on smooth domains, which the paper extends to arbitrary bounded open sets."},{"cited_title":"Contem- porary Math","cited_arxiv_id":null,"evidence_quote":"Provides the uniform lower bound argument for the nodal level used in Proposition 2.5 and in the compactness estimates."},{"cited_title":"6), Editor: M","cited_arxiv_id":null,"evidence_quote":"Supplies the exhaustion of an arbitrary bounded open set by smooth bounded open sets, used with Proposition 2.7 to transfer nodal ground states to rough domains."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that nodal action ground states are actual solutions of the equation, the bridge from Nehari sets to the PDE."},{"cited_title":"PDE 7(8) (2014), 1807–1838","cited_arxiv_id":null,"evidence_quote":"Gives the ball result on normalized supercritical solutions used as comparison and in Remark 1.6 for critical least-energy nodal solutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Pohozaev identity used on star-shaped domains to bound the energy of solutions from below in Theorem 1.7."}],"review_version":1}