{"id":"ba6a8346-d816-4de0-9bac-f1664b937a01","arxiv_id":"2411.17951","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For supercritical power combinations, the NLS with a small potential on large bounded domains admits a negative-energy minimizer and a positive-energy mountain pass solution; in the Sobolev critical case, a ground state and a high-energy solution exist for small mass.","lead":"This paper proves that a nonlinear Schrödinger equation on a bounded vessel has at least two standing wave solutions with a prescribed amount of mass, for certain combinations of focusing and defocusing nonlinear terms. A generalist reader might care because normalized solutions are the basic objects used to model Bose-Einstein condensates and light pulses, and the proof works on domains that need not be star-shaped.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.3's compactness step is not justified: the displayed inequality silently requires the gradient-smallness condition (22), absent from the theorem and from Theorem 1.2(i), and the equality preceding it is not derivable from (14)-(16).","rationale":"The reader's weakest-assumption list includes the misapplied Esteban-Lions half-space nonexistence and the unstated gradient-smallness condition. My stress-test identifies the Theorem 2.3 compactness step as the single load-bearing issue because it directly supports the positive-energy mountain-pass solution in Theorem 1.2(i), the paper's main novelty for non-star-shaped domains. The displayed computation after (14)-(16) is internally inconsistent: the equality claims a quantity is o(1) plus non-vanishing integrals, and the final negative bound uses condition (22), which is not assumed in Theorem 2.3. No Pohozaev identity on a bounded domain can supply the missing terms, so the proof of strong convergence is genuinely absent. The defect may be repairable: because the p-term enters the functional with a positive sign for β=-1, a direct Brezis-Lieb subtraction of the limit equation appears to force the gradient loss to zero without any gradient-smallness condition. That is why I recommend no change to the reader's conditional verdict rather than rejection: the results are plausible but the written proof of a central theorem has a real gap that must be fixed. The concrete test proposed would decide whether the proof needs a new hypothesis or just a correct compactness argument.","tokens_in":25925,"tokens_out":21403,"duration_ms":200867,"concrete_test":"Re-derive the compactness step in Theorem 2.3 by subtracting the weak-limit equation (16) from the PS equation (14) and applying Brezis-Lieb to ∫|u_n|^p. If this yields ‖∇(u_n-u_{r,s})‖_2 → 0 without invoking the gradient-smallness condition (22), then the gap is cosmetic and Theorem 1.2(i) survives as stated; if the convergence cannot be proved without adding (22), then Theorem 2.3 and Theorem 1.2(i) need a strengthened hypothesis or a fundamentally different compactness argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence claim for a positive-energy solution depends on Theorem 2.3, which asserts that a bounded Palais-Smale sequence for the modified problem (5) converges. After deriving (14)-(16), the proof displays an identity of the form o(1) = (gradient terms) + (q-2)/(2q)∫V u^2 - (p-q)/(pq)∫|u|^p, followed by a bound that is negative only if the unstated condition q‖Ṽ_+‖_{N/2} + N(q-2)‖V_+‖_{N/2} < S(2N-(N-2)q) holds. This condition appears nowhere in the hypotheses of Theorem 2.3 or in Theorem 1.2(i) except in the separate 'Moreover' clause. The two extra integrals in the displayed equality do not tend to zero, their signs are never explained, and no Pohozaev identity is available on the bounded domain Ω_r to justify them. Thus the strong convergence of the PS sequence is not established by the written argument. Since Theorem 2.5 passes s→1 using exactly this compactness, the existence of the mountain-pass solution ur with positive energy in Theorem 1.2(i) is not proven as it stands. The defect is potentially repairable, but the repair requires a genuine re-derivation, not a correction of notation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies normalized solutions of the NLS equation with a bounded potential V and the inhomogeneous nonlinearity |u|^{q-2}u + β|u|^{p-2}u on bounded domains Ω_r = rΩ. In the case q* < q < p ≤ 2* with β = -1, the authors claim, under an explicit smallness condition on V and for large mass a > a_V and large domains r, the existence of both a positive mountain-pass solution with positive energy and a positive global minimizer with negative energy, without assuming Ω is star-shaped. In the case 2 < q < q* < p = 2* with β = 1, they claim a local minimizer (ground state under extra assumptions) and a high-energy solution for small mass on star-shaped domains. The proofs use the monotonicity trick of Borthwick-Chang-Jeanjean-Soave, minimization on the sphere, and compactness analysis via blow-up and Liouville-type theorems.","tokens_in":26160,"tokens_out":7512,"duration_ms":67982,"significance":"If the results are correct, the paper makes a substantial contribution: it removes the star-shapedness assumption in the L2-supercritical combined-nonlinearity problem, which is an open problem raised by Bartsch, Qi and Zou, and it gives both positive- and negative-energy normalized solutions with explicit mass thresholds. The extension to the Sobolev-critical Brézis-Nirenberg regime for normalized ground states is also new. The strengths of the paper include explicit, parameter-free thresholds a_V and ~a_V, the use of an imported monotonicity trick rather than fitted parameters, and a clear presentation of the two different parameter regimes. However, the proof of the mountain-pass compactness in Theorem 2.3 contains a serious gap, and a key nonexistence assertion in Lemma 2.9 is not supported by the cited reference; these issues affect the central positive-energy existence claim of Theorem 1.2(i).","major_comments":[{"comment":"The compactness argument for the bounded Palais-Smale sequence is not valid as written. The displayed identity mixing integrals over Ω_r and R^N, with the extra terms (q-2)/(2q)∫_{R^N} V u_{r,s}^2 dx - (p-q)/(pq)∫_{R^N} |u_{r,s}|^p dx, does not follow from (14)-(16). The subsequent inequality requires the condition q||~V_+||_{N/2} + N(q-2)||V_+||_{N/2} < S(2N-(N-2)q), which is not assumed in the statement of Theorem 2.3 and appears only in the separate 'Moreover' clause of Theorem 1.2(i). Since Theorem 2.5 and hence the positive-energy solution in Theorem 1.2(i) depend on this compactness, the existence proof is incomplete as written; a genuine re-derivation of the strong convergence is needed.","section":"Theorem 2.3, paragraph following (16)"},{"comment":"The nonexistence assertion for the half-space limiting equation is not justified. Equation (25) is written as -Δu + λ∞u = |u|^{q-2}u - |u|^{q-2}u, which is evidently a typo; the intended equation is -Δu + λ∞u = |u|^{q-2}u - |u|^{p-2}u. The text attributes the nonexistence of nontrivial solutions to Esteban-Lions [14], but that reference treats pure power nonlinearities, not the combined sign-changing term with q ≠ p. Thus the conclusion lim inf_{r→∞} λr > 0 in Lemma 2.9, which is used in the 'Moreover' clause of Theorem 1.2(i), needs a different argument or a reference that actually covers this equation.","section":"Lemma 2.9, equation (25)"},{"comment":"The proof of strong convergence of the minimizing sequence invokes the identity I'_r(u_r) - λ_r u_r = 0 before u_r has been shown to be a critical point. At that stage u_r is only the weak limit of {u_n}, and the claim that the limit of the Lagrange multipliers λ_n corresponds to u_r is circular. The compactness can be obtained by testing the Palais-Smale condition I'_r(u_n) - λ_n u_n → 0 against u_n - u_r, which avoids assuming u_r is critical in advance; the proof should be rewritten accordingly.","section":"Theorem 3.1(ii), Lagrange multiplier passage"}],"minor_comments":[{"comment":"The last term '- s/p ∫_{Ω} |v_t|^p dx' duplicates the critical term '- s/2* ∫_{Ω} |v_t|^{2*} dx' because p = 2* in this section; this appears to be a typographical error.","section":"Lemma 3.2(i), displayed estimate for ~I_{1/t,s}(v_t)"},{"comment":"The reference to 'Theorem 2.4' at the start of the proof should be 'Lemma 2.4'.","section":"Lemma 2.9"},{"comment":"The phrase 'Liouville's theorem [14]' is used for the nonexistence of positive solutions of -Δv = |v|^{q-2}v on a half-space or R^N; this is a valid use for a pure power equation, but the precise statement and range of q from [14] should be quoted.","section":"Lemma 2.6"},{"comment":"There are multiple typos: 'Pohozave identity' (Lemma 2.9), 'direction calculation' (Lemma 2.9), 'moutain pass' (Theorem 3.5), and 'Arze-Ascoli' (Lemma 2.6). These should be corrected.","section":"Throughout"},{"comment":"The function u0,s is defined as v_{1/r*} and hence belongs to S_{r*,a}; for the conclusion stated for all r > r_a, the zero-extension of u0,s to Ω_r should be made explicit, since otherwise the mountain-pass endpoints are not literally in S_{r,a}.","section":"Lemma 2.1(i)"}],"recommendation":"major_revision","confidential_remarks":"The main claimed achievement — positive- and negative-energy normalized solutions on non-star-shaped domains for the L2-supercritical combined nonlinearity — is significant if it can be made rigorous. The gap in Theorem 2.3 is not a mere typo: the compactness step relies on an unstated gradient-smallness condition and on an identity that does not follow from the preceding equations. The misuse of the Esteban-Lions reference in Lemma 2.9 is also substantive. Both are potentially repairable within the manuscript's scope, so I recommend major revision rather than rejection. The authors should also reexamine the circular step in Theorem 3.1(ii), though there a standard fix exists."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know: the authors are after a genuine open problem, and the main theorems are likely true, but one load-bearing compactness proof is currently not justified. If repaired, this would be a real step forward: Theorem 1.2 removes the star-shaped hypothesis for the L2-supercritical combined nonlinearity on large bounded domains, and Theorem 1.4 gives a normalized ground state in the Sobolev-critical combined case under small mass. The global minimizer part (Theorem 2.8) is clean and essentially correct: the energy is bounded below even for q*<q<p because the Lq-term is controlled by the Lp-term at fixed mass, and no star-shapedness is needed.\n\nThe problems are in the high-energy constructions. In Theorem 2.3 the compactness argument for the Palais–Smale sequence is garbled. The displayed o(1) identity adds terms that do not follow from (14)–(16), and the negative upper bound silently uses the gradient-smallness condition (22), which is not part of Theorem 2.3’s assumptions (it appears later in Lemma 2.9). Since Theorem 2.5 passes s→1 using exactly this compactness, the existence of the positive-energy mountain pass solution in Theorem 1.2(i) is not proven as it stands. The defect is repairable but requires a genuine re-derivation, not a typo fix. Lemma 2.9 also leans on an Esteban–Lions nonexistence result for a half-space equation with sign-changing nonlinearity; that reference covers pure power nonlinearities and does not obviously apply, and the displayed equation (25) has a typo (the second nonlinearity is printed as |u|^{q−2}u rather than |u|^{p−2}u). Theorem 1.4(ii) uses boundedness of ∇V·x but only assumes V∈C^1; add the hypothesis or drop the claim. These are all fixable, but they are real weaknesses in the written proofs.\n\nThe paper is honest about what it does and does not do, the references look appropriate, and the main ideas are visible. It deserves a serious referee, and I would send it back for major revision, with specific requests: rewrite the compactness proof in Theorem 2.3, justify or replace the half-space nonexistence claim, and align the hypotheses in Theorem 1.4(ii). If those are satisfied, I would be happy to see it published.","headline":"A genuine open-problem paper where the main claims are likely true but the mountain-pass compactness argument in Theorem 2.3 has a real gap that needs repair before the positive-energy solution is proven.","tokens_in":26750,"tokens_out":4279,"would_cite":false,"duration_ms":35844,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J20","35J60","35Q55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves two positive normalized solutions for a focusing–defocusing NLS equation on large bounded domains of arbitrary shape, and a ground state plus a high-energy solution in the Sobolev-critical focusing case on star-shaped…","keywords":["NLS equations","normalized solutions","prescribed mass","bounded domains","L2-supercritical nonlinearity","Sobolev critical exponent","monotonicity trick","multiplicity"],"falsifier":"Look for a nonzero solution of $-\\Delta w+\\lambda w=|w|^{q-2}w-|w|^{p-2}w$ in the half-space $\\{x_1>0\\}$ with zero boundary condition, for some $N\\ge3$, $q^*<q<p\\le2^*$ and $\\lambda\\le0$; a numerical or variational construction of such a solution would invalidate the compactness step. Conversely, proving nonexistence for the sign-changing nonlinearity by the moving-plane or Pohozaev methods would close the gap.","tokens_in":25679,"feed_emoji":"🌀","tokens_out":7583,"duration_ms":63344,"temperature":0.7,"pith_summary":"The paper studies standing waves of the nonlinear Schrödinger equation with an external potential and an inhomogeneous combined nonlinearity $|u|^{q-2}u+\\beta|u|^{p-2}u$ on a bounded domain, with the mass $\\int|u|^2$ prescribed. It claims that when $2+\\frac4N<q<p\\le2^*$ and $\\beta=-1$, for every sufficiently large mass and every sufficiently large dilation of a fixed bounded domain, the problem has two positive solutions: a high-energy mountain pass solution and a negative-energy global minimizer. The main structural point is that this two-solution statement does not require the domain to be star-shaped, relaxing a hypothesis present in earlier work. For the complementary case $2<q<2+\\frac4N<p=2^*$ and $\\beta=1$, it claims a ground state and a high-energy solution on large star-shaped domains for small mass, a setting described as new for the Brézis-Nirenberg problem. A sympathetic reader would care because normalized solutions are the mass-preserving standing waves of the time-dependent equation, and bounded domains lack the scaling and translation invariances that make the whole-space theory work.","feed_headline":"Large bounded domains host two normalized NLS solutions","feed_subtitle":"For a focusing-defocusing nonlinearity, large domains of any shape give both a ground state and a high-energy solution with fixed mass.","key_machinery":"The argument is carried by three mechanisms: the monotonicity trick of [8], which produces bounded Palais–Smale sequences at almost every level of a parameter $s\\in[1/2,1]$; a rescaling $v_t(x)=t^{N/2}v_1(tx)$ of the first Dirichlet eigenfunction that converts the bounded domain into a large one and makes the energy functional track a one-variable function $h(t)$ whose sign change determines the mass threshold $a_V$; and Liouville/Pohozaev nonexistence on half-spaces and $\\mathbb R^N$ used to rule out concentration along the boundary or at infinity. In the Sobolev-critical case, the Aubin–Talenti bubbles $U_\\varepsilon$ and the ground state from the local minimization are used to estimate the mountain pass level below the first concentration threshold.","core_discovery":"The paper's central claim, stated as Theorem 1.2, is that for $N\\ge3$, $q^*<q<p\\le2^*$, $\\beta=-1$, and a bounded smooth domain $\\Omega$, under the assumption $(V0)$ on the potential, for every $a>a_V$ and every sufficiently large $r$, the constrained problem (4) has a positive mountain pass solution $u_r$ with $I_r(u_r)>0$ and a positive global minimizer $u_r$ with $I_r(u_r)<0$. Under an additional smallness condition involving $\\tilde V(x)=\\nabla V(x)\\cdot x$, the corresponding Lagrange multipliers satisfy $\\liminf_{r\\to\\infty}\\lambda_r>0$. The paper further claims, in Theorems 1.4 and 1.5, that for $2<q<q^*<p=2^*$ and $\\beta=1$, small mass on large star-shaped domains yields a local minimizer that is a ground state and, under boundedness of $\\tilde V$, a second high-energy solution.","pith_inferences":["If the half-space nonexistence step can be justified for the sign-changing nonlinearity (or replaced by another compactness argument), the same two-solution pattern should hold for a whole family of inhomogeneous nonlinearities whose energy is bounded below on the mass sphere; the paper's geometric two-well structure is the load-bearing part.","The dichotomy $a>a_V$ versus $a<\\tilde a_V$ is likely generic: when the highest-degree term is defocusing, large masses help create the two wells, whereas when it is focusing and critical, only small masses avoid concentration.","The existence of a global minimizer in a mass-supercritical regime looks like a bounded-domain effect: the volume term in the Gagliardo–Nirenberg inequality, together with the defocusing higher-power term, prevents the energy from escaping to $-\\infty$, something impossible on $\\mathbb R^N$."],"forward_implications":["For the focusing-defocusing range, large bounded domains of arbitrary shape carry two normalized positive solutions when the mass is large; this removes the star-shaped condition from earlier existence theorems.","The negative-energy solution is a global minimizer of the energy on the mass sphere, so a mass-supercritical combined nonlinearity can still admit a global minimizer on bounded domains.","In the Sobolev-critical focusing case, small mass on large star-shaped domains gives a ground state and a high-energy solution; the paper states this is new even when $V\\equiv0$ in the Brézis-Nirenberg context.","Under the extra potential-gradient smallness, the Lagrange multipliers of the high-energy and minimizer solutions stay bounded away from zero as the domain expands, so the frequencies do not drift to zero or infinity."],"supporting_citations":[{"why":"The open problem of normalized solutions on large bounded domains with potential and inhomogeneous nonlinearity; the paper's non-star-shaped result directly extends this work.","marker":"[4]"},{"why":"Supplies the monotonicity trick and the bounded Palais-Smale sequence theorem used to find critical points at almost every parameter.","marker":"[8]"},{"why":"The Liouville/nonexistence theorem on half-spaces invoked to rule out boundary concentration in the compactness arguments.","marker":"[14]"},{"why":"Introduced the normalized-solution variational framework with prescribed mass that the paper builds on.","marker":"[17]"},{"why":"First bounded-domain normalized results with local minimizer and mountain pass solution for L2-critical/supercritical pure powers on balls.","marker":"[21]"},{"why":"Extends local minimizer existence to generic bounded domains and gives the minimization technique used for the negative-energy solution.","marker":"[23]"},{"why":"Ground states for combined nonlinearities on the whole space; provides the Pohozaev-manifold and monotonicity toolbox.","marker":"[27]"},{"why":"Sobolev critical combined nonlinearities; source of level estimates and the mountain pass approach used in the critical case.","marker":"[28]"},{"why":"Two normalized solutions on star-shaped bounded domains for the Brézis-Nirenberg problem; supplies the Pohozaev identity and bubble estimates used in Section 3.","marker":"[29]"}],"fun_headline_variants":["Bounded domains host two normalized NLS states","Non-star-shaped domains get NLS ground states","Large mass, any shape: dual NLS solutions","Small mass on star-shaped domains yields two states","NLS ground and high-energy states on bounded domains"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of compactness for the mountain pass sequence requires that the limiting equation $-\\Delta w+\\lambda w=|w|^{q-2}w-|w|^{p-2}w$ on a half-space (and on $\\mathbb R^N$) has no nontrivial solution, and this is justified by a cited Liouville theorem that was proved only for pure power nonlinearities; a separate smallness condition on $\\nabla V$ is also used without being assumed in the statements.","fun_headline_variants_meta":{"raw":{"variants":["Bounded domains host two normalized NLS states","Non-star-shaped domains get NLS ground states","Large mass, any shape: dual NLS solutions","Small mass on star-shaped domains yields two states","NLS ground and high-energy states on bounded domains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1543,"prompt_tokens":1048,"completion_tokens":495,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":422}},"tokens_in":664,"tokens_out":495,"duration_ms":5041,"temperature":1.0,"reasoning_tokens":422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:45:29.226728+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a nonzero solution of $-\\Delta w+\\lambda w=|w|^{q-2}w-|w|^{p-2}w$ in the half-space $\\{x_1>0\\}$ with zero boundary condition, for some $N\\ge3$, $q^*<q<p\\le2^*$ and $\\lambda\\le0$; a numerical or variational construction of such a solution would invalidate the compactness step. Conversely, proving nonexistence for the sign-changing nonlinearity by the moving-plane or Pohozaev methods would close the gap.","supporting_citations":[{"cited_title":"Bartsch, S","cited_arxiv_id":null,"evidence_quote":"The open problem of normalized solutions on large bounded domains with potential and inhomogeneous nonlinearity; the paper's non-star-shaped result directly extends this work."},{"cited_title":"Borthwick, X","cited_arxiv_id":null,"evidence_quote":"Supplies the monotonicity trick and the bounded Palais-Smale sequence theorem used to find critical points at almost every parameter."},{"cited_title":"Esteban, P","cited_arxiv_id":null,"evidence_quote":"The Liouville/nonexistence theorem on half-spaces invoked to rule out boundary concentration in the compactness arguments."},{"cited_title":"Jeanjean, Existence of solutions with prescribed norm for s emilinear elliptic equations, Nonlinear Anal","cited_arxiv_id":null,"evidence_quote":"Introduced the normalized-solution variational framework with prescribed mass that the paper builds on."},{"cited_title":"Noris, H","cited_arxiv_id":null,"evidence_quote":"First bounded-domain normalized results with local minimizer and mountain pass solution for L2-critical/supercritical pure powers on balls."},{"cited_title":"Pierotti, G","cited_arxiv_id":null,"evidence_quote":"Extends local minimizer existence to generic bounded domains and gives the minimization technique used for the negative-energy solution."},{"cited_title":"Soave, Normalized ground states for the NLS equation with c ombined nonlinearities, J","cited_arxiv_id":null,"evidence_quote":"Ground states for combined nonlinearities on the whole space; provides the Pohozaev-manifold and monotonicity toolbox."},{"cited_title":"Soave, Normalized ground states for the NLS equation with c ombined nonlinearities: the Sobolev critical case, J","cited_arxiv_id":null,"evidence_quote":"Sobolev critical combined nonlinearities; source of level estimates and the mountain pass approach used in the critical case."}],"review_version":1}