{"id":"22c7fa4e-eecd-43f2-a198-6216aba498a4","arxiv_id":"2412.03296","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A variational existence proof for normalized critical NLS solutions on large domains contains a false Liouville step and sign errors, so the main theorems are not established.","lead":"This paper claims to prove existence and multiplicity of fixed-mass standing waves for a nonlinear Schrödinger equation with a critical nonlinearity and a potential, on large domains and in the whole space. The proofs contain several load-bearing mathematical errors and the claims are not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.3's uniform L∞ bound relies on a Liouville theorem that is false: the R^N blow-up limit admits the positive Aubin–Talenti bubble, so the claimed contradiction ω(0)=1 vs ω≡0 does not follow; Theorems 1.1(ii), 1.2(i), and 1.4(i) are unsupported as written.","rationale":"The reader identified the false Liouville step in Lemma 3.3 as a weakest assumption, and the full text confirms that this step is load-bearing. The paper's primary new conclusions include uniform L∞ bounds for solution families as r → ∞ and the resulting whole-space limits; Lemma 3.3 is the only argument supplying those bounds. The blow-up rescaling at a maximum point is standard, but the limiting equation in the R^N case is the critical Yamabe equation, which is known to possess positive solutions. Therefore the contradiction ω(0)=1 versus ω≡0 is not valid, and the proof of Theorem 1.1(ii) fails. The same lemma is invoked through Remark 3.4 for Theorems 1.2 and 1.4, so the uniform-bound portions of those theorems also lack support. There are additional concerns in the paper, including the treatment of V v_ε² terms in Lemmas 4.2 and 5.2 and a sign issue in Lemma 5.2 for β ≤ 0, but the Lemma 3.3 issue is the most central because it affects a common step in all three main theorems. My recommendation is unchanged: the manuscript should not be accepted in its current form, and a substantial revision is needed.","tokens_in":23535,"tokens_out":4388,"duration_ms":43723,"concrete_test":"Check whether the limit equation in Lemma 3.3 admits a positive solution in R^N by directly computing -ΔU_c for U_c(x) = c(1+|x|^2)^{-(N-2)/2} with c = (N(N-2))^{(N-2)/4}. A direct radial computation gives -ΔU_c = U_c^{2*-1}. If this substitution is verified, then the Liouville nonexistence assertion used to force ω ≡ 0 in Lemma 3.3 is false, and the claimed uniform L∞ bound cannot be obtained from that argument as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.3 is the load-bearing compactness step for the uniform L∞ estimates used in Theorems 1.1(ii), 1.2(i), and 1.4(i), and indirectly for the passage to whole-space limits. In the blow-up argument, after rescaling at a maximum point x_r with τ_r = M_r^{-2/(N-2)}, the potential and subcritical terms vanish, and any nontrivial limit ω satisfies -Δω = |ω|^{2*-2}ω on Σ. When the rescaled limit domain is Σ = R^N, the proof invokes a Liouville theorem to conclude ω ≡ 0, contradicting ω(0) = 1. There is no such Liouville theorem: the standard bubble U_c(x) = c(1+|x|^2)^{-(N-2)/2} with c = (N(N-2))^{(N-2)/4} is a positive smooth solution of exactly -Δω = ω^{2*-1} in R^N. Thus the contradiction does not follow, and the uniform L∞ bound for H^1-bounded families of solutions is not established. Since the paper relies on this bound to justify several asymptotic conclusions and the whole-space limit, the central claim is not supported without a replacement argument. This is an internal mathematical error, not merely a disagreement with an external convention. The earlier energy-based lower bound on λ_r in Theorem 3.2 does not repair this gap, because the uniform L∞ estimate is needed for the solution family itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies prescribed-mass (normalized) solutions to the nonlinear Schrödinger equation with a potential and combined subcritical/critical power nonlinearities, on large smooth star-shaped bounded domains Ω_r and then in the whole space R^N. The main results (Theorems 1.1, 1.2, 1.4) assert existence of local-minimum and mountain-pass type solutions for small mass, uniform L^∞ bounds as r→∞, and strictly positive Lagrange multipliers in the limit, with different signs of β. The proof strategy uses energy estimates on the constraint manifold, the monotonicity trick of Jeanjean, and a blow-up argument to rule out concentration. The central compactness mechanism is Lemma 3.3, which claims a uniform L^∞ bound by rescaling at a maximum point and invoking a Liouville theorem for the critical equation in R^N.","tokens_in":23752,"tokens_out":5237,"duration_ms":45211,"significance":"If correct, the paper would extend the recent work of Bartsch–Qi–Zou to the Sobolev critical case and would provide the first normalized-solution results with critical growth on large bounded domains with a nonconstant potential, including a passage to the whole space. The local-minimum existence for β>0 (Theorem 1.1(i)) appears to be a plausible and potentially useful contribution. However, the mountain-pass existence results and the uniform bounds that support the asymptotic and whole-space statements rest on several load-bearing estimates that contain mathematical errors. The false Liouville step and the sign error in the β≤0 estimate are not cosmetic; they invalidate the proofs of Theorems 1.2 and 1.4 as written. The significance of the paper would be high if the gaps could be repaired, but in its current form the central package is not established.","major_comments":[{"comment":"The uniform L^∞ bound is proved by a blow-up argument that invokes a Liouville theorem for the limit equation -Δω = |ω|^{2*-2}ω in Σ. When Σ = R^N, the text says 'from the Liouville theorem, there holds ω=0 in Σ,' contradicting ω(0)=1. This is false: the Aubin–Talenti bubble U_c(x) = c(1+|x|^2)^{-(N-2)/2}, with c=(N(N-2))^{(N-2)/4}, is a positive smooth solution of exactly that equation in R^N. Thus the claimed contradiction does not follow. This step is load-bearing for the uniform L^∞ bounds in Theorems 1.1(ii), 1.2(i), and 1.4(i), and indirectly for the passage to whole-space limits. The energy-based lower bound on λ_r in Theorem 3.2 does not repair this gap because the uniform bound is needed for the solution family itself.","section":"Section 3, Lemma 3.3"},{"comment":"In the estimate for m_{r,s}(α), the term -β/p T_1^{N(p-2)/2} ∫|v_ε|^p dx is nonnegative when β≤0, since -β ≥ 0. However, in Cases 1 and 2 the proof treats this term as a negative contribution and concludes that the whole expression is < 0. This is a sign error. Consequently the strict upper bound m_{r,s}(α) < (1/N) s^{(2-N)/2} S^{N/2} is not established, and the exclusion of bubble concentration in Theorem 5.3 fails.","section":"Section 5, Lemma 5.2"},{"comment":"The potential term (1/2)∫_Ω V(x) v_ε^2 dx is not higher-order in ε at the bubble scale. Since v_ε = (√α/||u_ε||_2) U_ε and ||U_ε||_2^2 = C ε^2 (for N≥5), while ∫ V U_ε^2 dx ~ V(0) ||U_ε||_2^2 when V(0)≠0, this term tends to (α/2)V(0) as ε→0, which is O(1), not O(ε^{N-2}). In Lemma 4.2 this term is absorbed into O(ε^{N-2}) without justification, and in Lemma 5.2 the attempted cancellation via the β term has the wrong sign (see the previous comment). The upper-bound estimates for the mountain-pass levels in Theorems 1.2 and 1.4 are therefore not justified.","section":"Section 4, Lemma 4.2 and Section 5, Lemma 5.2"}],"minor_comments":[{"comment":"There are numerous typos and notation issues, e.g., 'Pincaré' should be 'Poincaré', 'Brezis-Lieb' should be 'Brézis-Lieb', and the Abstract contains 'Besides, Our study' with an improper capital letter.","section":"Throughout"},{"comment":"The statement of the monotonicity trick contains a garbled line '֒→֒→E′' that appears to be a formatting error; the intended embedding E into H should be stated correctly.","section":"Section 2, Theorem 2.2"},{"comment":"The proof states 'lim inf_{r→∞} dist(x_r, ∂Ω_r)/τ_r > 0' and says it follows from a standard direct method ('So we omit it here'). Since this claim is used to identify the limit domain Σ, it should be either proved or given a precise reference.","section":"Section 3, Lemma 3.3"},{"comment":"The proof refers to 'the proof of (iii)' but the Lemma has no item (iii); this appears to be a remnant from an earlier version and should be corrected.","section":"Section 5, Lemma 5.2"},{"comment":"After defining the path γ(τ), the proof uses γ0(t) and max_{t≥0} J_{r,s}(v_t); the notation for the path and the variable t should be made consistent.","section":"Section 4, Lemma 4.2"}],"recommendation":"reject","confidential_remarks":"The local-minimum part for β>0 (Theorem 1.1(i)) may survive the errors because its proof does not rely on the false Liouville step, but the mountain-pass results and the uniform L^∞ bounds are central to the paper's stated contribution. The false Liouville theorem, the sign error in the β≤0 estimate, and the mishandling of the potential at bubble scale are load-bearing and cannot be repaired by local adjustments within the manuscript's scope. The paper would require a fundamentally different compactness argument or substantial additional assumptions (e.g., V(0)≤0 or a different monotonicity mechanism) to support the claimed results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first attack on the Sobolev-critical normalized problem with a nonconstant potential on large domains, and that is a real gap in the literature. But the central compactness machinery has three load-bearing errors. I would not trust the main theorems as written.\n\nWhat is new and good: the paper correctly identifies the Bartsch–Qi–Zou subcritical result and extends it to q = 2*, which is not a restatement. The local-minimum existence for β > 0 and small mass (Theorem 1.1(i)) is plausibly correct: the lower bound on E_r on B_{r,α}, the Ekeland argument, and the energy gap excluding a bubble are standard and the signs there check out. The overall variational setup is honest and the exposition is readable.\n\nWhere it breaks: Lemma 3.3 is the load-bearing uniform L^∞ estimate. The blow-up rescaling at a maximum leads to a limit solving -Δω = |ω|^{2*-2}ω on R^N. The proof invokes a Liouville theorem to get ω ≡ 0 and a contradiction with ω(0) = 1. No such theorem exists: the Aubin–Talenti bubble is a positive solution. So the contradiction fails, and every result that relies on uniform L^∞ (parts (ii) of 1.1, (i) of 1.2, (i) of 1.4, whole-space limits) is unsupported. The lower bound on λ_r in Theorem 3.2 does not repair this.\n\nLemma 5.2 has a sign error. For β ≤ 0 the term -β/p ∫|v_ε|^p is nonnegative, but the proof treats it as a negative contribution and concludes the mountain-pass level is below S^{N/2}/N. Also the potential term ∫ V v_ε^2 is O(1) (roughly α V(0)) for continuous V, not O(ε^{N-2}), so the upper bound in Lemma 5.2 (and Lemma 4.2) is not valid. For N = 3 the proof is not even completed. These are not cosmetic gaps; they are exactly the estimates needed to rule out bubbles in the mountain-pass concentration argument.\n\nNet: the claimed theorems are not established. The β > 0 local minimum part may survive, and the question itself deserves attention. But the paper needs a different compactness argument, not copy-editing. I would still send it to a serious referee—this is a known-hard problem and the authors are aiming at the right target—but I would expect heavy revision or a substantially weakened set of results.","headline":"Genuinely novel critical-case program, but the compactness proof leans on a false Liouville step and a sign error, so the main existence claims do not hold as written.","tokens_in":24329,"tokens_out":4161,"would_cite":false,"duration_ms":41104,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J60","35J20","35R25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that a nonlinear Schrödinger equation with an external potential and both subcritical and Sobolev-critical power nonlinearities admits solutions with a prescribed L² mass on large star-shaped domains, and that these…","keywords":["Normalized solutions","prescribed mass","Sobolev critical exponent","combined nonlinearities","potential","large smooth domains","mountain pass","compactness"],"falsifier":"Evaluate Lemma 3.3's Liouville step: for the blow-up limit Σ=R^N, the lemma claims the only nonnegative solution of -Δω = $ω^{{2*-1}}$ is ω≡0, but substituting the explicit bubble U_ε (a positive solution with ω(0)=1 after rescaling) into equation (3.6) with V≡0 gives a valid nonzero limit, so the lemma's conclusion cannot be true; hence the uniform L∞ bound, and with it the whole-space passage, is not established by this proof.","tokens_in":23217,"feed_emoji":"📐","tokens_out":9346,"duration_ms":80768,"temperature":0.7,"pith_summary":"This paper establishes that a nonlinear Schrödinger equation with an external potential and both subcritical and Sobolev-critical power nonlinearities admits solutions with a prescribed L² mass on large star-shaped domains, and that these solutions persist as the domain expands to all of R^N. Because the potential makes the usual Pohozaev-manifold method unavailable, the authors work directly on the mass sphere and control the energy by an explicit mass threshold: for small mass the energy stays below the critical bubble level, which restores compactness. For β>0 they find a positive local-minimum solution with negative energy; for β>0 and for β≤0 they find positive mountain-pass solutions with positive energy. The solution families are uniformly bounded in L∞ and have Lagrange multiplier λ bounded away from zero as the radius tends to infinity, which is the step that yields whole-space solutions.","feed_headline":"Small mass keeps compactness in critical Schrödinger equation","feed_subtitle":"Below a mass threshold, local-minimum and mountain-pass solutions exist and survive as the domain grows to R^N.","key_machinery":"The central object is the mass-threshold function h(t) (and its descendants f, g) that bounds the constrained energy from below in terms of ‖∇u‖². The threshold α_V is chosen so that h has a positive global maximum with two roots R10, the energy functional restricted to the mass sphere has a local minimum inside the sublevel ball B_{r,α} = {‖∇u‖²≤T_α²}, giving a positive solution with negative energy; crossing from one side of the mountain yields a second, positive-energy solution. For β≤0 the same crossing argument works with the critical term alone driving the mountain pass. What makes the critical exponent tractable is the observation that any loss of compactness costs at least $S^{{N/2}}$/N in energy; the thresholds α_V, ᾱ_V, α1, α2 are chosen so that all energy levels considered lie strictly below that cost, so Palais–Smale sequences converge strongly. Expanding r→∞ then gives normalized solutions on all of R^N.","pith_inferences":["Because the blow-up analysis ignores the potential at the concentration scale, a natural testable extension is whether the uniform bounds persist for non-vanishing potentials, e.g., V(0)≠0; the scaling in Lemmas 4.2 and 5.2 suggests they may need a corrected potential term or a stronger smallness condition.","The threshold structure suggests a possible dichotomy: for masses above α_V, either no normalized solution exists or the energy reaches the bubble level and compactness is lost, so solutions would concentrate and the whole-space limit would fail; checking the autonomous case V≡0 against Soave's results could reveal whether the thresholds are sharp or merely sufficient.","The same energy-below-bubble-threshold strategy could transfer to normalized solutions for systems, to magnetic Schrödinger operators, or to problems on domains with different symmetries, wherever the Pohozaev constraint is unavailable."],"forward_implications":["If the theorems hold, equation (1.1) on R^N has positive normalized solutions of both local-minimum and mountain-pass type for every sufficiently small mass α, obtained as limits of the domain solutions as r→∞.","The explicit thresholds (α_V, ᾱ_V, α1, α2) express the smallness condition on the mass in terms of V, p, N, and the Sobolev constant, giving a quantitative answer to the open problem raised in Bartsch et al.","The uniform L∞ bounds and liminf λ>0 mean these whole-space limits are genuine localized waves, not bubbling peaks escaping to infinity.","For β≤0 the mountain-pass result extends the known autonomous case to potentials, showing the focusing critical term can be stabilized by small mass even when the subcritical term is defocusing.","The method avoids the Pohozaev manifold entirely, so it can be applied to bounded domains and to potentials where the Pohozaev identity gives no constraint."],"supporting_citations":[{"why":"The work this paper extends; introduced the large-domain and mass-threshold strategy for combined subcritical/supercritical nonlinearities with potential, which the critical case must complement.","marker":"[7]"},{"why":"Established ground states and mountain-pass solutions for the same equation with V=0 on R^N in the Sobolev critical case; supplies the critical energy level S^{N/2}/N that controls compactness here.","marker":"[35]"},{"why":"Classic existence of normalized solutions via Pohozaev and mountain pass; the paper explicitly notes this approach fails with a potential, motivating the new constrained minimization.","marker":"[23]"},{"why":"Source of the bubble estimates in Lemma 2.3 and of the min-max geometry used in the mountain-pass sections (Lemma 7.1 cited there).","marker":"[25]"},{"why":"Defines the optimal Sobolev constant S and its embedding inequality, used throughout the energy thresholds and bubble threshold.","marker":"[4]"},{"why":"Provides the sharp Gagliardo-Nirenberg inequality that bounds the subcritical term in the energy lower estimates.","marker":"[40]"},{"why":"The monotonicity trick (Theorem 2.2) used to obtain bounded Palais-Smale sequences at almost every level for the perturbed functionals.","marker":"[11,15]"},{"why":"Elliptic Lp estimates and boundary regularity used in the blow-up analysis (Lemma 3.3) to pass to the limit equation.","marker":"[20]"}],"fun_headline_variants":["Mass below threshold yields two critical Schrödinger solutions","Critical Schrödinger: small mass sustains solutions as domain grows","Small mass unlocks Schrödinger solutions on large domains","Two solutions below mass threshold in critical Schrödinger"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The uniform bounds that make the r→∞ limit work assume that the rescaled blow-up limit has no positive solution in R^N (or in a half-space) and that the potential term vanishes at the concentration scale; if the potential does not vanish at the blow-up point, that assumption fails and the boundedness estimates can collapse.","fun_headline_variants_meta":{"raw":{"variants":["Mass below threshold yields two critical Schrödinger solutions","Critical Schrödinger: small mass sustains solutions as domain grows","Small mass unlocks Schrödinger solutions on large domains","Two solutions below mass threshold in critical Schrödinger"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0008,"raw_usage":{"total_tokens":3491,"prompt_tokens":891,"completion_tokens":2600,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":2538}},"tokens_in":507,"tokens_out":2600,"duration_ms":18668,"temperature":1.0,"reasoning_tokens":2538,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:35:32.416607+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate Lemma 3.3's Liouville step: for the blow-up limit Σ=R^N, the lemma claims the only nonnegative solution of -Δω = $ω^{{2*-1}}$ is ω≡0, but substituting the explicit bubble U_ε (a positive solution with ω(0)=1 after rescaling) into equation (3.6) with V≡0 gives a valid nonzero limit, so the lemma's conclusion cannot be true; hence the uniform L∞ bound, and with it the whole-space passage, is not established by this proof.","supporting_citations":[{"cited_title":"(French)","cited_arxiv_id":null,"evidence_quote":"Defines the optimal Sobolev constant S and its embedding inequality, used throughout the energy thresholds and bubble threshold."}],"review_version":1}