{"id":"8d62089b-43a1-465b-b6f9-25bea1321a33","arxiv_id":"2501.04893","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For mass-supercritical nonlinearities with an external potential, positive normalized solutions exist on sufficiently large star-shaped domains and, under a radial condition on the potential, in R^N.","lead":"This paper proves existence of prescribed-mass solutions to a nonlinear Schrödinger equation with an external potential and a general mass-supercritical nonlinearity, first on large star-shaped domains and then in the whole space by taking the domain radius to infinity. The result extends known normalized-solution theory and complements a 2024 theorem by Bartsch-Qi-Zou.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1 scales trial functions by t^{3/2} instead of t^{N/2}, so v_t is only mass-admissible for N=3; the mountain-pass geometry and hence Theorem 1.1(i) are not established for the stated N≥3.","rationale":"The reader flagged condition (V1) as the weakest assumption and noted the scaling inconsistency in passing. I judge the scaling inconsistency more load-bearing: it affects existence on bounded domains for all c, before V1 is even used. Theorem 1.1(i) is built directly on Lemma 3.1; if the scaling exponent is wrong, the mountain-pass geometry collapses for N≠3. I agree with the reader that V1 is a strong condition and that the proof of Theorem 1.2 collapses without it, but that concern is secondary to the domain-level result being dimensionally correct. The scaling issue is concrete and testable, and it can be resolved by re-deriving the lemma with t^{N/2}. I do not call for rejection: the error may be a typographical/OCR issue, and the proof architecture is standard. My recommendation remains CONDITIONAL, with the added explicit condition that Lemma 3.1 must be redone with the correct dimension-dependent rescaling before the main theorems are accepted.","tokens_in":20716,"tokens_out":10056,"duration_ms":98894,"concrete_test":"Independently recompute Lemma 3.1 with the mass-correct rescaling v_t(x)=t^{N/2}v_1(tx). Verify that: (a) ∫|v_t|^2=c and ∫|∇v_t|^2=t^2θc; (b) the nonlinear term is controlled with exponent N(d'-2)/2 in place of 3(d'-2)/2; (c) the resulting h(t) has a zero and a negative tail for all N≥3 using α,β>2+4/N. If (a)-(c) hold, the concern is typographical or cosmetic; if not, Theorem 1.1(i) fails for those N because the min-max path is inadmissible.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3, Lemma 3.1: set v_t(x)=t^{3/2}v_1(tx), where v_1∈S_{1,c}. Then ∫_{Ω_{1/t}}|v_t|^2 dx = t^{3-N}c, so v_t∉S_{1,c} for N≠3. The subsequent estimates (3.2)-(3.7) inherit this: the exponent 3(d'-2)/2 should be N(d'-2)/2, and the path γ_0 in part (iii) is not a path in S_{r,c} unless N=3. Because Theorem 3.2's monotonicity trick and Lemma 3.5(i) use exactly this geometry, the existence of (λ_{r,c},u_{r,c}) in Theorem 1.1(i) is unproved for N≠3 as the manuscript stands. This is an internal correctness risk, not a disagreement with consensus. If the '3' is an OCR artifact for 'N', the proof can likely be repaired; if not, the stated theorem is only dimensionally valid for N=3.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies normalized solutions (fixed L^2 mass) of the nonlinear Schrödinger equation -Δu+V(x)u+λu=g(u) on large star-shaped bounded domains Ω_r and on R^N, under mass-supercritical growth assumptions on the general nonlinearity g. The authors use the monotonicity trick and a mountain-pass geometry on the mass constraint manifold to prove, for every c>0, existence of a positive mountain-pass type solution on Ω_r for r sufficiently large; under an additional small-mass condition and a strong exponential decay/growth condition (V1), they obtain a positive solution on R^N with λ>0. The paper is framed as a complement to a recent work of Bartsch-Qi-Zou.","tokens_in":20907,"tokens_out":14340,"duration_ms":113571,"significance":"If the results are fully established, they constitute a useful extension of the normalized-solution theory to nonconstant potentials and general mass-supercritical nonlinearities on large domains, including a passage to the whole space with positive Lagrange multiplier. The proof strategy is standard and the paper contains no fitted free parameters; the claims are concrete and falsifiable. However, the current proof contains a dimensional-scaling error that, as written, restricts the main existence result to N=3, so the significance for the stated N≥3 depends on a repair of that argument.","major_comments":[{"comment":"The trial function v_t(x)=t^{3/2}v_1(tx) is mass-preserving only in dimension N=3. Indeed, ∫_{Ω_{1/t}}|v_t|^2 dx = t^{3-N}c, so v_t∉S_{1,c} for N≠3. Because the estimates (3.2)–(3.7), the choice of endpoints u_0 and u_1, and the path γ_0 in part (iii) all use the same 3/2 scaling and the exponent 3(d'-2)/2 in the definition of h(t), the mountain-pass geometry of Lemma 3.1 is established only for N=3. Since Theorem 3.2 and Lemma 3.5(i) rely on this geometry, Theorem 1.1(i) is not proved for the stated N≥3. The proof can likely be repaired by replacing 3 with N throughout, but as written the argument covers only the three-dimensional case.","section":"Section 3, Lemma 3.1"},{"comment":"The final assertion 'lim_{c→∞} E_V(u_c)=∞' is inconsistent with the quantifier 'for any 0<c<\\tilde c' preceding it, and Section 4 proves only the passage r→∞ for fixed c; no proof of an energy limit in c appears anywhere. If the intended limit is c→0+, it still requires an argument; if not, the assertion should be removed or corrected.","section":"Theorem 1.2"},{"comment":"The statement 'It follows from (3.15) that m0>0' is not justified: (3.15) concerns lim inf of max_{Ω_r} u_{r,c}, whereas z^1_r is a concentration center of the remainder ν^1_r=u_r-u_0, and there is no evident relation between the two. The half-space exclusion may be obtainable directly for m0=0 as well, but the manuscript needs to supply the missing reasoning or clarify the role of (3.15).","section":"Section 4, Lemma 4.1, Step 2"}],"minor_comments":[{"comment":"The abstract advertises 'multiplicity' and 'bifurcation property', but no theorem in the manuscript concerns multiplicity or bifurcation; the abstract should be aligned with the actual results.","section":"Abstract"},{"comment":"In the proof of (ii), Case 2, the phrase 'by an argument similar to that in Lemma 3.5' is a self-reference; presumably another lemma (e.g., Lemma 3.4) is meant, and the citation should be corrected.","section":"Lemma 3.5"},{"comment":"The phrase 'for any ϕ∈C_c^∞(Ω_r)' should read 'for any ϕ∈C_c^∞(Σ)' to make the limiting argument meaningful.","section":"Section 4, before (4.5)"},{"comment":"The citation '[20, Theorem 9.11]' for L^p estimates should be the Gilbarg-Trudinger reference [21], since [20] is a different paper.","section":"Lemma 3.4"},{"comment":"The sentence 'Since E_{r,s}(u1)≤0 for any γ∈Γ_{r,c}, we have ...' is garbled; it should be phrased as 'for every γ∈Γ_{r,c}, the path must cross the level set...' or similar.","section":"Lemma 3.1(iii)"}],"recommendation":"major_revision","confidential_remarks":"The scaling exponent issue is almost certainly a mechanical '3' for 'N' substitution, but it pervades Lemma 3.1 and currently invalidates the main theorem for N≠3. The theorem's energy-limit assertion is also unsupported. The paper is within the journal's scope and the overall strategy is sound; I recommend major revision with a request for a thorough audit of all exponents and cross-references."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a plausible within-field extension: it takes the standard normalized NLS machinery (monotonicity trick, Pohozaev identity, blow-up analysis, concentration compactness) and pushes it to large star-shaped domains with a general potential and mass-supercritical nonlinearity, then passes to the whole space as r→∞. That's a legitimate complement to Bartsch-Qi-Zou and Ding-Zhong, and the positioning looks honest.\n\nThe main problem is in Lemma 3.1. The trial function is v_t(x)=t^{3/2}v_1(tx), which only lies on the mass sphere if N=3. The exponent should be N/2. This error propagates into the estimates: h(t) and (3.2) use t^{3(d'-2)/2} instead of t^{N(d'-2)/2}, and (3.5) has t^{3(d-2)/4} instead of t^{N(d-2)/4}. Since the mountain-pass geometry in Lemma 3.1(iii) depends on this scaling, Theorem 1.1(i) is not established for N≠3 as written. The fix is straightforward if '3' is a typo for 'N' — and the presence of N elsewhere in the formulas suggests it is — but it has to be done.\n\nThere are also smaller mechanical issues: Theorem 1.2 states \"lim_{c→∞} E_V(u_c)=∞\" where it should be c→0+; the abstract promises a bifurcation result that never appears; and Lemma 3.5 contains a self-reference ('by an argument similar to that in Lemma 3.5') that should point to a different lemma.\n\nOn the substance, the strategy is standard and the estimates are mostly routine. Condition (V1), the exponential lower bound on the radial derivative of V, is doing a lot of work in ruling out bubbles moving off to infinity. It is strong and somewhat artificial, but that's a sufficient-condition issue, not a fatal one.\n\nOverall, this is solid within-field progress, likely correct after the scaling typo is fixed. It doesn't open new territory, but it fills a real gap. I'd send it to a competent referee — the scaling error is exactly the kind of thing a referee should catch — but I wouldn't accept it until the Lemma 3.1 issue is resolved. If the authors confirm the N/2 fix, the paper deserves publication.","headline":"Plausible within-field extension of normalized NLS results to large domains with a potential, but Lemma 3.1's scaling is wrong for N≠3 and needs fixing before the main theorem holds.","tokens_in":21462,"tokens_out":5998,"would_cite":false,"duration_ms":51368,"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 proves that the mass-constrained nonlinear Schrödinger equation with a nonconstant potential admits positive normalized mountain-pass solutions on every sufficiently large star-shaped domain, and, for small masses, on all of…","keywords":["Normalized solutions","Mass supercritical","Nonconstant potential","Large smooth domains","Star-shaped domain","Mountain pass solution","Concentration compactness","Nonlinear Schrödinger equation"],"falsifier":"Fix $N=3$, take $V(x)=-\\varepsilon(1+|x|^2)^{-1}$ and $g(u)=|u|^{p-2}u$ with $4<p<6$, and solve the constrained problem on $B_r$ for $c=1$ by a numerical mountain-pass algorithm for a sequence $r\\to\\infty$. The theorem predicts a positive solution with positive energy and $\\lambda_r$ eventually positive for small $c$; finding a large $r$ with no such critical point, or a branch with $\\lambda_r\\to -\\infty$ while $\\|u_r\\|_{\\infty}$ stays bounded, would refute it.","tokens_in":20491,"feed_emoji":"🌊","tokens_out":9575,"duration_ms":94005,"temperature":0.7,"pith_summary":"This paper establishes existence of solutions with prescribed $L^2$ mass for the nonlinear Schrödinger equation $-\\Delta u+V(x)u+\\lambda u=g(u)$ when the nonlinearity is mass-supercritical and $V$ is nonconstant. Because the potential breaks the Pohozaev-manifold reduction used in the autonomous case, the authors work directly on the mass sphere in large star-shaped domains, using the monotonicity trick and a Pohozaev-type identity adapted to the potential. They prove that for every $c>0$ there is a radius $r_c$ such that the problem on every rescaled domain $\\Omega_r$ with $r>r_c$ has a positive mountain-pass solution with positive energy. Under a quantitative smallness condition on the positive part of $x\\cdot\\nabla V$, they then take $r\\to\\infty$ and obtain a positive normalized solution on the whole space for small masses, with positive Lagrange multiplier. If correct, this supplies the missing mass-supercritical, nonconstant-potential case for large bounded domains and its whole-space limit.","feed_headline":"Prescribed-mass waves exist on every large star-shaped domain","feed_subtitle":"With a nonconstant potential, even small prescribed masses yield a whole-space solution branch.","key_machinery":"The carrying structure is the family of truncated energy functionals $E_{r,s}(u)=\\frac12\\int_{\\Omega_r}|\\nabla u|^2+\\frac12\\int_{\\Omega_r} V u^2-s\\int_{\\Omega_r}G(u)$ on the mass sphere $S_{r,c}$, together with the monotonicity trick that produces bounded Palais-Smale sequences for almost every $s$. A Pohozaev-type identity, combined with the star-shaped geometry ($x\\cdot n\\ge 0$ on the boundary), converts the mountain-pass level into a uniform $H^1$ bound independent of $r$ and $s$. As $r\\to\\infty$, a concentration-compactness decomposition separates the whole-space solution $u_0$ from finitely many translated bubbles $w_k(\\cdot-z_r^k)$ centered at points escaping to infinity; condition (V1), the exponential lower bound on $x\\cdot\\nabla V$ along rays, is precisely what rules those bubbles out, yielding strong convergence.","core_discovery":"The central claim, stated on the paper's own terms, is Theorem 1.1 and Theorem 1.2: for every $c>0$ there exists $r_c>0$ such that problem (1.1)--(1.2) on $\\Omega_r$ possesses a positive mountain-pass type solution $(\\lambda_{r,c},u_{r,c})$ with $E_V(u_{r,c})>0$, and the family obeys a uniform $L^\\infty$ bound as $r\\to\\infty$. If additionally $\\|\\tilde V_+\\|_{N/2}<2S$ with $\\tilde V(x)=x\\cdot\\nabla V(x)$, then for $0<c<\\tilde c$ the frequencies satisfy $\\liminf_{r\\to\\infty}\\lambda_{r,c}>0$, and passing to the limit gives a positive solution $(\\lambda_c,u_c)$ on $\\mathbb{R}^N$ with $\\lambda_c>0$, $E_V(u_c)>0$, and $E_V(u_c)\\to\\infty$ as $c\\to\\infty$. The proof establishes this by showing that no bubbles can escape to infinity during the domain expansion, so the $r\\to\\infty$ limit is strong rather than a superposition of bubbles.","pith_inferences":["If condition (V1) is weakened to polynomial decay of $x\\cdot\\nabla V$, the exponential-weight argument that kills the bubbles fails at a specific surface-integral estimate; a plausible consequence is that the $r\\to\\infty$ limit consists of a whole-space solution plus one or more bubbles at infinity, so the domain-size limit would be a superposition rather than a single solution.","The quantitative threshold $\\|\\tilde V_+\\|_{N/2}<2S$ suggests a concrete test: compute the largest $c$ for which the whole-space solution's frequency stays positive and compare it with the constant $\\tilde c$ that would follow from optimizing the Gagliardo-Nirenberg constants in the proof.","The same machinery should extend to coupled systems or potentials with singularities, as long as the mass-supercritical two-sided growth bounds and the exponential radial-growth condition hold; that extension is not stated in the paper.","One can try to construct an explicit potential $V(x)=-\\varepsilon(1+|x|^2)^{-1}$ and a pure power $g(u)=|u|^{p-2}u$ with $2+4/N<p<2^*$, and solve the constrained problem numerically on large balls; matching the predicted mountain-pass level and positive $\\lambda_r$ would test the quantitative constants, while observing bubble concentration would suggest the exponential condition is needed."],"forward_implications":["For any prescribed mass $c>0$, every sufficiently large star-shaped domain carries a positive normalized solution of mountain-pass type with positive energy, even though the energy is unbounded below on the mass sphere.","The uniform $L^\\infty$ bound as $r\\to\\infty$ means the family of domain solutions does not develop spikes at the boundary; the whole-space solution is approached in a controlled way.","Under the smallness condition $\\|\\tilde V_+\\|_{N/2}<2S$, small masses yield whole-space solutions with positive frequency $\\lambda_c>0$, so they behave like bound states rather than zero-frequency limits.","The solution branch persists for all large masses in the whole space, with energy $E_V(u_c)\\to\\infty$ as $c\\to\\infty$.","The result covers general mass-supercritical nonlinearities satisfying two-sided power bounds, not just pure power nonlinearities."],"supporting_citations":[{"why":"Supplies the large-domain framework and the bubble-decomposition limit that the present work extends to general mass-supercritical nonlinearities.","marker":"[7]"},{"why":"Provides the monotonicity trick that yields bounded Palais-Smale sequences on the mass sphere.","marker":"[24]"},{"why":"Gives the Gagliardo-Nirenberg inequality with sharp constants used for the energy lower bounds.","marker":"[42]"},{"why":"Ruled-out half-space solutions that would arise from bubbles approaching the boundary in the $r\\to\\infty$ limit.","marker":"[19]"},{"why":"Classification of nonnegative solutions of the limiting autonomous equation used to exclude bubbles in the whole space when $\\beta<2^*$.","marker":"[16]"},{"why":"Interior and boundary elliptic estimates used in the uniform $L^\\infty$ bound and the exponential decay estimates for bubbles.","marker":"[21]"}],"fun_headline_variants":["Mass supercritical waves exist on every large star-shaped domain","Potential barrier bypassed: normalized waves on expanding domains","Strong limit from bounded to whole-space for prescribed-mass waves","Mountain-pass solutions with potential on large smooth star-shaped domains"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the radial derivative $x\\cdot\\nabla V(x)$ stays positive and decays slower than every exponential as $|x|\\to\\infty$; if a potential satisfies the other hypotheses but violates this exponential growth condition, the proof's mechanism for excluding bubbles escaping to infinity no longer works, and the whole-space conclusion may fail.","fun_headline_variants_meta":{"raw":{"variants":["Mass supercritical waves exist on every large star-shaped domain","Potential barrier bypassed: normalized waves on expanding domains","Strong limit from bounded to whole-space for prescribed-mass waves","Mountain-pass solutions with potential on large smooth star-shaped domains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"total_tokens":1293,"prompt_tokens":1000,"completion_tokens":293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":226}},"tokens_in":616,"tokens_out":293,"duration_ms":4067,"temperature":1.0,"reasoning_tokens":226,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:05:49.014242+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix $N=3$, take $V(x)=-\\varepsilon(1+|x|^2)^{-1}$ and $g(u)=|u|^{p-2}u$ with $4<p<6$, and solve the constrained problem on $B_r$ for $c=1$ by a numerical mountain-pass algorithm for a sequence $r\\to\\infty$. The theorem predicts a positive solution with positive energy and $\\lambda_r$ eventually positive for small $c$; finding a large $r$ with no such critical point, or a branch with $\\lambda_r\\to -\\infty$ while $\\|u_r\\|_{\\infty}$ stays bounded, would refute it.","supporting_citations":[{"cited_title":"Nonlinear Anal","cited_arxiv_id":null,"evidence_quote":"Provides the monotonicity trick that yields bounded Palais-Smale sequences on the mass sphere."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Gagliardo-Nirenberg inequality with sharp constants used for the energy lower bounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Ruled-out half-space solutions that would arise from bubbles approaching the boundary in the $r\\to\\infty$ limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classification of nonnegative solutions of the limiting autonomous equation used to exclude bubbles in the whole space when $\\beta<2^*$."},{"cited_title":"Springer, Berlin (1983)","cited_arxiv_id":null,"evidence_quote":"Interior and boundary elliptic estimates used in the uniform $L^\\infty$ bound and the exponential decay estimates for bubbles."}],"review_version":1}