{"id":"e4ce1eda-f46e-4b91-bc5d-199be86ba649","arxiv_id":"2507.23639","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For N≥6, the normalized Sobolev-critical NLS with potential has a positive mountain-pass solution; for N≥3, a negative-energy local minimizer exists under weaker conditions than in the cited preprint.","lead":"This paper proves that a nonlinear Schrödinger equation with a prescribed mass and a critical nonlinearity has a 'mountain-pass' solution in dimensions six and higher, under suitable attractive potential assumptions. It also shows a negative-energy local minimizer exists in all dimensions N≥3 with weaker assumptions than a recent preprint.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.3's estimate (41) is not justified by condition (8) alone: the positive parts V1^+ and V2^+ are not controlled, so ∫V Uε^2 can exceed -C||Uε||_2^2 and the mountain-pass level bound can fail.","rationale":"The reader flagged condition (8) and Lemma 3.3, and I agree those are load-bearing. Digging into the proof, the specific failure point is (41), where condition (8) is used as if it made the whole potential negative on the support of Uε. It only controls V1^- against V2^-; the positive parts V1^+,V2^+ are unrestricted except for the vague constants C2,C3 in (7), which are introduced for the ground-state argument and never connected to the size of the bubble correction. Since V1 and W1 are only in L^{N/2} and L^N, integrable singularities are allowed, and such singularities can make ∫V^+Uε^2 of order ε^{2-α}, which dominates ||Uε||_2^2∼ε^2 for any α>0. This is not a matter of style: inequality (41) is the only place in Lemma 3.3 where the attractive-potential assumption enters, and without a corrected estimate the mountain-pass level cannot be pushed under αµ+(1/N)S^{N/2}μ^{1-N/2}. The proposed test settles whether (41) is salvageable under the current assumptions. I therefore keep the reader's CONDITIONAL verdict: the central claim may be true after adding a quantitative smallness condition on the positive parts of V, but the proof as written lacks it.","tokens_in":13,"tokens_out":21963,"duration_ms":391835,"concrete_test":"Set N=6, R*=1, V1^-=Mχ_{B_1}, V1^+=κ|x|^{-3/2}χ_{B_1}, V2=-mχ_{B_L} with L large, M>m, κ small so (3)-(8) hold. Compute Iε=∫V Uε^2. Scaling gives Iε=-c1 M ε^2 + c2 κ ε^{1/2}+o(ε^{1/2}); since ||Uε||_2^2∼c0 ε^2, Iε/||Uε||_2^2→∞. Equation (41) requires Iε≤-C||Uε||_2^2, so it fails. If an independent lemma proved ∫V1^+Uε^2=o(ε^2) under (3)-(8), the concern would be resolved; no such lemma appears.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.3, Step 2, is the decisive step for Theorem 1.4. In (39) the minimax value is bounded by sup_t G(t), where G(t)=t^2/2 S^{N/2}-μt^{2*}/2* S^{N/2}+t^2/2||V2^-||∞||Uε||_2^2+t^2/2∫V Uε^2 dx. The proof then says that by condition (8), G(t)<...-C t^2||Uε||_2^2, i.e. it uses ∫V Uε^2 dx ≤ -C'||Uε||_2^2. Condition (8) compares V1^- with ||V2^-||∞ pointwise; it says nothing about V1^+ or V2^+. For an admissible V1^+(x)=κ|x|^{-α}χ_{B_1}(0) with 0<α<2 and N≥6, direct scaling gives ∫V1^+Uε^2 = c κ ε^{2-α}+o(ε^{2-α}), while ||Uε||_2^2 = c1ε^2+O(ε^{N-2}); thus ∫V^+Uε^2/||Uε||_2^2 →∞. Assumption (7) only requires ||V1^+||_{N/2}<C2 and a^2||V2^+||∞<C3 with C2,C3 chosen in Lemma 2.6 for the ground-state argument, not for this estimate; nothing links them to the gap in (8). Choosing V1^-=Mχ_{B_{2R*}}, V2^-=-mχ_{B_L} with M>m and κ small satisfies (3)-(8) but makes ∫V Uε^2 positive of order ε^{2-α}, so (41) is false as stated. Without (41), the strict inequality θµ<αµ+(1/N)S^{N/2}μ^{1-N/2} is not established, and the mountain-pass solution argument collapses unless a new smallness condition on V^+ is added.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies normalized solutions of the NLS equation -Δu+V(x)u=λu+|u|^{2*-2}u on R^N with prescribed L^2 norm a^2, where V=V1+V2 with V1∈L^{N/2} and V2∈L^∞ decaying at infinity. Under hypotheses (3)-(7), Theorem 1.2 establishes a local minimizer/ground state with negative energy for all N≥3; under the additional condition (8), inf_{B_{2R*}} V1^- > ||V2^-||∞, Theorem 1.4 claims a mountain-pass solution with positive energy for N≥6, which would close an open problem in Verzini-Yu [19, Remark 1.11]. The proof normalizes the mass to one, introduces the ground-state level αμ, constructs a path from the ground state v̄ to a high-energy configuration, and estimates the minimax level by comparing a concentrated bubble Uε with the critical bubble energy (1/N)S^{N/2}μ^{1-N/2}.","tokens_in":13628,"tokens_out":13345,"duration_ms":135733,"significance":"If the results were fully supported, Theorem 1.4 would solve the N≥6 case left open in [19] and Theorem 1.2 would modestly improve the local-minimizer result in the same paper. The variational strategy—Pohozaev identities, a positive-energy barrier, a local minimizer, and a bubble-based mountain-pass comparison—is a natural and potentially valuable approach. However, the central energy comparison in Lemma 3.3 is not justified by condition (8), and several load-bearing lemmas are deferred to [19] without details. The main theorem therefore needs substantive revision before the claim can be accepted.","major_comments":[{"comment":"Condition (8) does not imply the estimate claimed in Eq. (41). Since V = V1^+ - V1^- + V2^+ - V2^-, condition (8) controls only V1^- on B_{2R*}; it does not control V1^+ or V2^+. The proof silently uses ∫ V Uε^2 dx ≤ -C||Uε||_2^2, but with V1^+(x)=κ|x|^{-α}χ_{B_1}(0), 0<α<2 and N≥6, scaling gives ∫ V1^+ Uε^2 = cκ ε^{2-α}+o(ε^{2-α}), while ||Uε||_2^2 = c1 ε^2+O(ε^{N-2}). Thus the positive contribution can dominate and ∫ V Uε^2 can be positive of order ε^{2-α}. Such a potential can satisfy (3)-(8): take V1^-=Mχ_{B_{2R*}} and V2=-mχ_{B_L} with M>m and κ small. Therefore Eq. (41) is false as stated, and the strict inequality θμ<αμ+(1/N)S^{N/2}μ^{1-N/2} is not established. A smallness condition on V^+ relative to the gap in (8), or a different estimate, is needed.","section":"Section 3, Lemma 3.3, Eq. (41)"},{"comment":"The attainment of αμ is not proved; the text says only \"Similar to the argument of [19, Lemma 4.2], we can prove that αμ < 0 is achieved.\" Since the minimizer v̄ at level αμ is the base point of the minimax path in Section 3 and is needed to define Γ, this is a load-bearing step. The authors should either provide the proof in full or state precisely which hypotheses of [19, Lemma 4.2] apply and why the same argument works here.","section":"Section 2, Lemma 2.5"},{"comment":"The existence of a bounded Palais-Smale sequence at level θμ and the strong convergence to a nonzero critical point are deferred to '[19, Lemma 5.6]' with no details. In particular, excluding the alternative in Lemma 2.3(ii) relies on the strict gap θμ<αμ+(1/N)S^{N/2}μ^{1-N/2}; once Lemma 3.3's gap is invalid, this convergence argument loses its foundation. A self-contained proof of Lemma 3.5 is required.","section":"Section 3, Lemma 3.5"}],"minor_comments":[{"comment":"The condition \"1 - t||Uε||_2^2 > 0\" should read \"1 - t^2||Uε||_2^2 > 0\" to match the definition of ψε,t = (1 - t^2||Uε||_2^2)^{1/2} v̄ + tUε.","section":"Section 3, Lemma 3.3, Step 1"},{"comment":"The notation V1^- and V2^- is used in condition (8) before the positive/negative part notation is defined; it should be introduced explicitly.","section":"Introduction, condition (8)"},{"comment":"Lemma 3.1 is applied with φ = v̄, but Lemma 3.1 requires φ ∈ L^∞_{loc}(B_{2R*}); the local boundedness of the ground state v̄ is not established in the paper.","section":"Section 3, proof of Lemma 3.3"},{"comment":"The sentence \"Similar to the arguments of [19, Lemmas 4.1 and 4.2]\" covers the existence and convergence of the minimizing sequence; these arguments should be written out or precisely located in [19] to make the paper self-contained.","section":"Section 2, proof of Theorem 1.2"},{"comment":"There are several typographical issues, such as \"correspondis\" in Theorem 1.2 and \"weaking\" in the introduction; these should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is not supported by the proof as it stands: Lemma 3.3's key estimate (41) does not follow from condition (8), and the counterexample with a singular positive part V1^+ shows that the claimed negative shift in the bubble energy can be overwhelmed. The manuscript also leans very heavily on deferred arguments from [19] and [16] for the attainment of αμ and for the Palais-Smale analysis. The result may be repairable by adding a genuine smallness condition on V^+ and by supplying the missing compactness arguments, but the present version is not ready for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: the claimed solution of the N≥6 open problem from [19, Remark 1.11] is not established by the proof as written. The gap is in Lemma 3.3, and the stress-test concern lands exactly there. What is actually new: Theorem 1.2 weakens the hypotheses for the negative-energy local minimizer in [19], allowing the small-sphere radius to be controlled by V and avoiding separate V+ restrictions for N=3,4. That part looks plausible: the variational machinery is standard, and the deferred arguments in Lemma 2.5 are 'similar to' genuinely similar results. The local-minimizer theorem is a modest but real contribution.\n\nThe problem is Theorem 1.4. In Lemma 3.3, Step 2, condition (8) is used to deduce (41), i.e. that ∫ V Uε^2 ≤ -C ||Uε||_2^2. Condition (8) only control the negative parts V1^- and V2^- locally; V1^+ and V2^+ are free. Take V1^+(x)=κ|x|^{-α}χ_{B_1} with 0<α<2 and N≥6. This satisfies the L^{N/2} and W1 assumptions (and can have arbitrarily small L^{N/2} norm by taking κ small), but ∫ V1^+ Uε^2 ~ κ ε^{2-α}, while ||Uε||_2^2 ~ ε^2. For any fixed κ>0, the positive term dominates as ε→0. So (41) is false for admissible potentials. An explicit example: V1^- = Mχ_{B_{2R*}}, V2^- = -mχ_{B_L} with M>m, plus this V1^+, satisfies (3)–(8) for suitable parameters and makes ∫ V Uε^2 positive of order ε^{2-α}. The strict minimax bound θμ < αμ + (1/N)S^{N/2}μ^{1-N/2} collapses.\n\nThis is load-bearing, not a presentation issue. The mountain-pass theorem may be recoverable with an additional smallness or boundedness condition on V^+ near the concentration point, but that is not in the paper. The local-minimizer part may well be correct, and the paper seriously engages with the literature, so it deserves a real referee. Send it, but the referee should focus on Lemma 3.3 and ask for either a repaired estimate or an explicit extra assumption on V^+.\n\nBest.","headline":"The local-minimizer result looks plausible, but the N≥6 mountain-pass theorem has a load-bearing gap: condition (8) does not control V^+, so the key estimate (41) in Lemma 3.3 is unjustified.","tokens_in":34,"tokens_out":8803,"would_cite":false,"duration_ms":140211,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J20","35J60","35Q55","35B33"],"pacs":[],"model":"deepseek-v4-flash","headline":"For N≥6, the purely Sobolev-critical NLS with an attractive potential admits a positive normalized mountain-pass solution.","keywords":["normalized solutions","NLS equation","Sobolev critical exponent","mountain-pass solution","local minimizer","constrained critical points","Pohozaev identity","trapping potential"],"falsifier":"A concrete check is to evaluate the coefficient of $\\|U_\\varepsilon\\|_2^2$ in the mountain-pass level estimate (41) when condition (8) is reduced to equality, i.e. $\\inf_{B_{2R_*}} V_1^- = \\|V_2^-\\|_\\infty$. If the next-order terms still give $\\theta_\\mu < \\alpha_\\mu + \\frac{1}{N}S^{N/2}\\mu^{1-N/2}$, condition (8) is not sharp; if the inequality degenerates and a Palais–Smale sequence at that level can only converge up to a bubble, the strict form of (8) is necessary.","tokens_in":12929,"feed_emoji":"🧮","tokens_out":14594,"duration_ms":127562,"temperature":0.7,"pith_summary":"The paper studies the nonlinear Schrödinger equation with prescribed mass $\\int u^2 = a^2$, a potential $V$, and the purely Sobolev-critical nonlinearity $|u|^{2^*-2}u$, where $2^* = 2N/(N-2)$. It proves that under smallness conditions on the negative parts of $V$, a local minimizer with negative energy exists for every $N \\ge 3$, improving the assumptions of the earlier treatment of the same problem. Its main result is a mountain-pass solution with positive energy for $N \\ge 6$, the dimension range left open there. The new ingredient is a local-attractiveness condition on the potential: inside a fixed ball, the negative part of the localized component must strictly dominate the bounded component's negative part, which pushes the minimax level below the threshold where bubbling would break compactness. If correct, the two theorems complete the normalized-solution picture for this equation in all dimensions $N \\ge 3$.","feed_headline":"Critical NLS gets a mountain-pass solution for N≥6","feed_subtitle":"A local-attractiveness condition pushes the energy below the bubbling threshold, proving the missing mountain-pass solution for N≥6.","key_machinery":"The argument rescales the problem to the unit sphere by setting $v = u/a$ and $\\mu = a^{4/(N-2)}$, so solutions become critical points of $J_\\mu(v) = \\frac{1}{2}\\int |\\nabla v|^2 + \\frac{1}{2}\\int V v^2 - \\frac{\\mu}{2^*}\\int |v|^{2^*}$ on $M = \\{ \\int v^2 = 1 \\}$. Two objects carry the proof: the negative-energy local minimizer $\\bar{v}$ from Theorem 1.2, obtained by minimizing inside a gradient ball of radius $t_\\mu$, and the Sobolev bubble $U_\\varepsilon$, a cutoff of the standard optimizer of the Sobolev inequality used to test the energy. The paper builds test paths $\\Psi_{\\varepsilon,t} = \\xi^{(N-2)/2}((1 - t^2\\|U_\\varepsilon\\|_2^2)^{1/2}\\bar{v} + tU_\\varepsilon)(\\xi x)$, renormalized onto $M$, and estimates their energy along a further dilation $s \\mapsto (\\Psi_{\\varepsilon,t})_s$. Condition (8) makes the $\\varepsilon^{(N-2)/2}$-coefficient of the bubble term negative, yielding $\\theta_\\mu < \\alpha_\\mu + \\frac{1}{N}S^{N/2}\\mu^{1-N/2}$; a min-max principle for Hilbert manifolds then produces a bounded Palais–Smale sequence that converges strongly to the mountain-pass solution.","core_discovery":"The paper's central discovery is Theorem 1.4: for $N \\ge 6$, under assumptions (3)–(7) plus condition (8), problem (1) has a mountain-pass solution $\\bar{v}^* \\in H^1(\\mathbb{R}^N)$ with positive energy, a negative Lagrange multiplier, and $\\bar{v}^* > 0$ almost everywhere. It also proves Theorem 1.2: for all $N \\ge 3$, under (3)–(5), a local minimizer with negative energy exists, and under the additional smallness bounds (6) or (7) this minimizer is a ground state. Together these results remove the dimensional restriction in the previous treatment, where the mountain-pass solution was only available for $3 \\le N \\le 5$, and weaken the assumptions under which the negative-energy local minimizer is known to be a ground state. The mountain-pass solution is found by a minimax over paths on the $L^2$ sphere joining the local minimizer to a high-energy point, with the level controlled below the first bubbling threshold.","pith_inferences":["Condition (8) is probably sufficient rather than necessary: the proof only needs the $\\varepsilon^{(N-2)/2}$ coefficient in (41) to be negative, so an averaged or weighted form of the same local dominance could drive the identical minimax argument.","Because the ball radius $R_*$ is tied to the bubble cutoff, shrinking $R_*$ is itself part of the mechanism; the same strategy should extend to potentials with several attractive wells, producing multi-bump mountain-pass solutions.","A natural next question is dynamical: the mountain-pass solution sits above a local minimizer and is therefore a candidate for orbital instability under the associated time-dependent NLS flow, but the paper does not address stability."],"forward_implications":["For $N \\ge 6$, a sufficiently attractive potential yields at least two normalized solutions: the negative-energy local minimizer and the positive-energy mountain-pass solution.","The negative-energy minimizer is a ground state under the smallness conditions (6) for $N = 3,4$ and (7) for $N \\ge 5$, so the prescribed mass and the potential together determine the least-energy bound state.","The potential family $V(x) = -C/(1+|x|^\\tau)$ with $\\tau > 2$ satisfies all hypotheses for suitably small $C$, large cutoff radii, or small mass $a$, so the theorems apply to explicit trapping wells.","The strict level bound $\\theta_\\mu < \\alpha_\\mu + \\frac{1}{N}S^{N/2}\\mu^{1-N/2}$ blocks bubble loss of compactness, so the Palais–Smale sequence converges strongly instead of splitting off a Sobolev bubble.","The mountain-pass solution carries a negative Lagrange multiplier, so it is a bound state in the usual spectral sense even though its energy level is positive."],"supporting_citations":[{"why":"Supplies the preceding local minimizer and low-dimension mountain-pass results, the open $N \\ge 6$ problem, and several lemmas reused directly here.","marker":"[19]"},{"why":"Gives the pointwise-convergence relation used to obtain the Sobolev-bubble asymptotics (22).","marker":"[3]"},{"why":"Provides the abstract min-max principle that turns the level estimate into a convergent constrained critical sequence.","marker":"[5]"},{"why":"Supplies Lemma 5.5, the integral estimates for the cutoff bubble against bounded test functions, used in the mountain-pass level computation.","marker":"[16]"},{"why":"Provides Lemma 2.1, the Sobolev-space bounds on potential integrals, together with the two-solution framework for negative potentials.","marker":"[10]"},{"why":"Establishes the mountain-pass variational mechanism for normalized solutions that the path class is built to emulate.","marker":"[7]"}],"fun_headline_variants":["Mountain-pass solution found for critical NLS with potential for N≥6","Sobolev-critical NLS: mountain-pass solution for N≥6 solves open case","Critical NLS with potential: mountain-pass exists for N≥6","For N≥6 critical NLS yields mountain-pass solution","Higher-dimensional critical NLS: mountain-pass solution proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the potential to be strictly more attractive in a fixed ball than its bounded component's worst negative part anywhere; if that strict inequality becomes an equality, the energy estimate at the minimax level loses its margin and the argument no longer produces a strongly converging sequence.","fun_headline_variants_meta":{"raw":{"variants":["Mountain-pass solution found for critical NLS with potential for N≥6","Sobolev-critical NLS: mountain-pass solution for N≥6 solves open case","Critical NLS with potential: mountain-pass exists for N≥6","For N≥6 critical NLS yields mountain-pass solution","Higher-dimensional critical NLS: mountain-pass solution proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000783,"raw_usage":{"total_tokens":3416,"prompt_tokens":865,"completion_tokens":2551,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":2458}},"tokens_in":481,"tokens_out":2551,"duration_ms":18413,"temperature":1.0,"reasoning_tokens":2458,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:31:19.236603+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to evaluate the coefficient of $\\|U_\\varepsilon\\|_2^2$ in the mountain-pass level estimate (41) when condition (8) is reduced to equality, i.e. $\\inf_{B_{2R_*}} V_1^- = \\|V_2^-\\|_\\infty$. If the next-order terms still give $\\theta_\\mu < \\alpha_\\mu + \\frac{1}{N}S^{N/2}\\mu^{1-N/2}$, condition (8) is not sharp; if the inequality degenerates and a Palais–Smale sequence at that level can only converge up to a bubble, the strict form of (8) is necessary.","supporting_citations":[{"cited_title":"Normalized solutions for the nonlinear Schr\\\"odinger equation with potential: the purely Sobolev critical case","cited_arxiv_id":"2505.05357","evidence_quote":"Supplies the preceding local minimizer and low-dimension mountain-pass results, the open $N \\ge 6$ problem, and several lemmas reused directly here."},{"cited_title":"Br´ ezis, E.H","cited_arxiv_id":null,"evidence_quote":"Gives the pointwise-convergence relation used to obtain the Sobolev-bubble asymptotics (22)."},{"cited_title":"Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge Uni- versity Press, Cambridge, 1993","cited_arxiv_id":null,"evidence_quote":"Provides the abstract min-max principle that turns the level estimate into a convergent constrained critical sequence."},{"cited_title":"Molle, G","cited_arxiv_id":null,"evidence_quote":"Provides Lemma 2.1, the Sobolev-space bounds on potential integrals, together with the two-solution framework for negative potentials."},{"cited_title":"Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal","cited_arxiv_id":null,"evidence_quote":"Establishes the mountain-pass variational mechanism for normalized solutions that the path class is built to emulate."}],"review_version":1}