{"id":"eecf0579-0743-4128-b946-ef86e95e6f7b","arxiv_id":"2507.04624","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For sufficiently small prescribed mass, normalized solutions exist, and often many of them exist, for a large class of elliptic boundary value problems with general nonlinearities and several boundary condition types.","lead":"This paper proves existence and multiplicity of normalized solutions, solutions with a prescribed L2 mass, for a broad class of elliptic Schrodinger equations on bounded domains with Dirichlet, Neumann, or nonlinear Robin boundary conditions. The perturbation method it uses avoids scaling and Morse theory, so it works for continuous nonlinearities and opens cases, such as Neumann multiplicity, where no prior result existed.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1 (Robin problem) is false as stated because u=0 satisfies E'(0)=0 and E(0)≤M μ**; the proof of Theorem 1.5 needs a mass-positive formulation.","rationale":"The reader's weakest assumption (Lemma 2.5) survives scrutiny: the uniform bound, the Gagliardo-Nirenberg estimate, and the exponent bookkeeping all work, so the Dirichlet existence and multiplicity core appears sound. The most load-bearing unresolved issue is in the Robin section, where Lemma 3.1 as stated is false and its proof has exponent errors; since Theorem 1.5 depends on ruling out case (ii) via this lemma, the Robin existence claim is not rigorously established until the lemma is corrected. This is an internal inconsistency, not a disagreement with consensus, and it is concrete. The sign error in Theorem 5.3 and the unproved Section 6 theorems are additional flaws but they do not affect the Dirichlet/Robin/Neumann existence as directly as Lemma 3.1 affects the Robin theorem. Verdict remains conditional pending revision.","tokens_in":35504,"tokens_out":41468,"duration_ms":397322,"concrete_test":"Rewrite Lemma 3.1 with the additional hypothesis u≠0 (or ∫Ω u^2 dx = ν > 0), and redo the proof using ∥u∥^{p-2} ≤ (2qM μ**/(q-2))^{(p-2)/2} and ∥u∥^{l-2} ≤ (2qM μ**/(q-2))^{(l-2)/2}. Then check whether the resulting μ** still excludes all such low-energy unconstrained critical points with 0<ν<μ**; if yes, Theorem 1.5 is repairable, if no, the Robin existence claim lacks a proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.1 in Section 3 asserts that for small μ** there is no u∈H^1(Ω) with E'(u)=0 and E(u)≤M μ**. Since (f4) and (g2) imply f(0)=g(0)=0, u=0 is always such a critical point; the lemma is literally false. The proof divides by ∥u∥^2 and silently assumes u≠0. The needed statement is the Robin analogue of Dirichlet Lemma 2.5: no u≠0 with ∫u^2=ν>0, E'(u)=0, E(u)≤M μ** for 0<ν<μ**. This is exactly what is required to rule out case (ii) in the Robin version of Lemma 2.4, so Theorem 1.5 is not proven as written. The displayed inequality in the proof also uses ∥u∥^{p-2} ≤ (2qM μ**/(q-2))^{p-2}; the correct exponent is (p−2)/2, and similarly for l. The existence of μ** still follows after the correction, but the lemma must be restated and reproved. The Dirichlet Lemma 2.5, by contrast, includes ν>0 and its exponent bookkeeping checks out.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a perturbation method, in the style of Buffoni–Esteban–Séré, for normalized solutions of semilinear elliptic problems with prescribed L2 mass on bounded domains under Dirichlet, Robin, and Neumann boundary conditions. The main Dirichlet results (Theorems 1.1, 1.2, 1.4) assert existence and multiplicity of solutions (u,λ) with λ in a low-energy interval, for sufficiently small mass μ, under either (f1)–(f3) or (f1)+(f4), with an oddness assumption for multiplicity. Analogous Robin results (Theorems 1.5, 1.6) and Neumann multiplicity (Theorem 1.7) are claimed, together with a ground-state theorem (Theorem 5.3) and several further applications to exponential-critical, magnetic, biharmonic, and Choquard equations. The core Dirichlet argument is mostly written out: a penalized functional on a truncated mass ball, compactness of Cerami sequences, a nonexistence lemma ruling out mass loss, and a mountain-pass/Fountain scheme.","tokens_in":35713,"tokens_out":8867,"duration_ms":91047,"significance":"If the stated results are correct, the paper would provide the first multiplicity theorems for normalized Neumann solutions in this generality and would extend normalized-solution theory on bounded domains to general nonlinearities with no differentiability assumptions. The paper also has useful expository content: the appendix gives a self-contained proof of the mountain-pass theorem for C1 functionals on open sets, and the explicit μ-thresholds in Theorem 1.2 and Corollary 1.3 are concrete. However, the Robin section and the ground-state section currently contain load-bearing errors, so the significance of the contribution depends on repairs that are local but not merely cosmetic.","major_comments":[{"comment":"Lemma 3.1 is false as stated. Under (f4) and (g2) we have f(0)=g(0)=0, hence u=0 satisfies E'(0)=0 and E(0)=0≤M μ**, so the lemma's conclusion that no such u exists is contradicted by u=0. The proof divides by ||u||^2 and implicitly assumes u≠0. The statement that is actually needed for Theorem 1.5 is the Robin analogue of Lemma 2.5: for small ν>0 there is no nonzero critical point with ∫u^2=ν and E(u)≤M μ**. The current statement must be revised and the proof must treat u=0 separately; as written, Theorem 1.5 is not proven.","section":"Section 3, Lemma 3.1"},{"comment":"The displayed estimates in the proof of Lemma 3.1 contain incorrect exponents. From ||u||^2 ≤ 2qM μ**/(q−2) one obtains ||u||^{p−2} ≤ (2qM μ**/(q−2))^{(p−2)/2}, not the displayed (2qM μ**/(q−2))^{p−2}; likewise for the exponent l−2 instead of l−2, and the coefficient C'_{p,Ω} should be C'_{l,Ω}. The existence of a small μ** can still be recovered after correcting the exponents, but the inequalities as written are invalid.","section":"Section 3, proof of Lemma 3.1"},{"comment":"The quantity s := 2λ_{1,0,0}(q−2*)/(2*(q−2−4/N)) is not necessarily positive under the stated assumption q>2+4/N. For example, when N=3, q=4 satisfies q>2+4/N but q<2*=6, so the numerator is negative while the denominator is positive; the same possibility occurs for other N. The proof of Lemma 5.4 only yields the claimed lower bound for λ_u when q>2*, because the coefficient (1/2* − 1/q) must be nonnegative. When q<2*, the derived bound is negative and cannot support the definition of s as an equivalent-norm shift. Theorem 5.3 therefore needs an additional hypothesis such as q>2*, or a different argument, before the ground-state conclusion is valid.","section":"Section 5, Lemma 5.4 and Theorem 5.3"}],"minor_comments":[{"comment":"The problem displayed at the beginning of Section 4 is labeled (P)^μ_{1,0,0}, but the boundary condition shown is the Neumann condition, so the label should be (P)^μ_{0,1,0}.","section":"Section 4"},{"comment":"Theorem 6.1 states the solution has λ∈[0,λ_{1,1,1}], but for the Dirichlet problem in Ω⊂R^2 the relevant eigenvalue is λ_{1,0,0}; the subscript appears to be a typo.","section":"Section 6.1"},{"comment":"Theorems 6.3 and 6.4 refer to problem (6.1), but the magnetic-field problem is (6.2).","section":"Section 6.2"},{"comment":"There are several small typographical issues: 'the domain the domain' in the introduction, integrals written as ∫_Ω f(x,u) dx in Section 2 even though f is independent of x, and the notation (f1') containing f(x,t) with no x-dependence. These should be cleaned up.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The Dirichlet part of the paper seems largely coherent and the nonexistence Lemma 2.5 is not circular. The Robin section, however, contains a literally false lemma and exponent errors, and Theorem 5.3 has a sign issue in the definition of s. These are repairable in principle, so I do not recommend rejection, but the current version cannot be accepted until Lemma 3.1 is restated with a mass condition and proved correctly, and the ground-state theorem is given a valid positivity assumption or proof. The applications in Section 6 are stated without proofs and would need fuller verification in any revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: the paper has a genuinely useful idea—adapting the Buffoni–Esteban–Séré penalization to get normalized solutions on bounded domains under Dirichlet, Neumann, and Robin conditions—and the Dirichlet part (Theorems 1.1, 1.2, 1.4) looks essentially sound. The perturbation functional, the nonexistence Lemma 2.5, and the mountain pass appendix all check out. The multiplicity results for Neumann solutions are new as far as I can tell.\n\nBut there are two soft spots that need attention before the whole package is credible.\n\nFirst, Lemma 3.1 (the Robin nonexistence lemma) is false as stated. It claims no u with E'(u)=0 and E(u) ≤ M μ**, but u=0 satisfies exactly that. The proof divides by ||u||^2 and silently assumes u≠0; the statement needs a mass condition ν>0, as the Dirichlet Lemma 2.5 has. The exponent bookkeeping in the proof is also off: from ||u||^2 ≤ C μ** you get ||u||^{p−2} ≤ (C μ**)^{(p−2)/2}, not (C μ**)^{p−2}, and similarly for l. The lemma can be repaired, but as written Theorem 1.5 has a gap because case (ii) of the Robin version of Lemma 2.4 is not ruled out.\n\nSecond, Theorem 5.3 and its proof define s := 2λ1(q−2*)/(2*(q−2−4/N)) and call it positive. Under the paper's own assumptions, (f1) and (f4) imply q ≤ p < 2*, so q−2* is negative and s is negative. The lower bound on λu derived in Lemma 5.4 is therefore trivial (negative) and cannot support the shift argument. This section needs reworking, not just a sign fix.\n\nSection 6 also states eight theorems with no proofs. As sketches of future applications that is acceptable, but they should be labeled as such.\n\nNet: the core Dirichlet/Neumann existence results are a real contribution and the paper deserves serious reviewing, but it needs careful revision. I would send it to a referee with instructions to focus on Lemma 3.1 and Section 5.","headline":"Genuinely useful perturbation-method core for Dirichlet/Neumann, but Lemma 3.1 and Theorem 5.3 have fixable errors that need a thorough revision.","tokens_in":36301,"tokens_out":5630,"would_cite":false,"duration_ms":52647,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","35J25","35Q55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any sufficiently small prescribed L² mass, normalized solutions exist on bounded domains—and odd nonlinearities give arbitrarily many.","keywords":["Bounded domain","Normalized solutions","General Neumann boundary condition","Multiplicity","Perturbation method","Nonlinear Schrödinger equation","Prescribed L2 norm"],"falsifier":"To settle whether the central claim is right, look for a solution of $-\\Delta u=f(u)$ in $H^1_0(\\Omega)$ with $E(u)\\le\\lambda_{1,0,0}/2$ and $\\int_\\Omega u^2\\,dx=\\nu$ for some $\\nu<\\mu_0$: a single such critical point under (f1)–(f3) contradicts Lemma 2.5. Concretely, for $f(t)=|t|^{p-2}t$ with $2+4/N<p<2^*$ on the unit ball, one can check numerically whether branches of small-mass, low-energy solutions exist for arbitrarily small $\\nu$; the theorem predicts they do not.","tokens_in":35260,"feed_emoji":"📐","tokens_out":14181,"duration_ms":157631,"temperature":0.7,"pith_summary":"This paper establishes a small-mass threshold theorem for nonlinear Schrödinger-type equations on smooth bounded domains with general boundary conditions. If the nonlinearity $f$ is continuous and satisfies either the growth-plus-superlinearity conditions (f1)–(f3) or the growth condition together with the Ambrosetti–Rabinowitz condition (f1)+(f4), then for every sufficiently small prescribed mass $\\mu$ the Dirichlet problem $-\\Delta u=\\lambda u+f(u)$, $u\\in H^1_0(\\Omega)$, $\\int_\\Omega u^2\\,dx=\\mu$ has a solution with $\\lambda\\in[0,\\lambda_{1,0,0}]$. Under oddness of $f$, the same argument yields at least $m$ distinct solutions for any chosen $m$ once $\\mu$ is small enough. The method transfers to the Robin problem and, modulo the two constant solutions, to the Neumann problem; the paper states these are the first multiplicity results for normalized Neumann solutions in this setting. Because the proof uses a penalized unconstrained functional rather than scaling or Pohozaev-type identities, it also covers exponential-critical growth in $\\mathbb{R}^2$, magnetic fields, biharmonic equations, and Choquard equations.","feed_headline":"Small L2 mass guarantees solutions on bounded domains","feed_subtitle":"For merely continuous nonlinearities, a small prescribed norm gives existence—and oddness yields arbitrarily many.","key_machinery":"The machine is the penalized functional $E_{r,\\mu}$ on the open set $U_\\mu$, together with the barrier $H_{r,\\mu}(u)=(\\int_\\Omega|u|^2\\,dx/\\mu)^r/(1-\\int_\\Omega|u|^2\\,dx/\\mu)$. As $r$ grows, the barrier uniformly vanishes on compact subsets of $U_\\mu$ but still prevents Cerami sequences from crossing the mass sphere, and the truncated functional $J_{r,\\mu}$ is $C^1$ on the whole space. The companion result that carries the argument is Lemma 2.5, a nonexistence statement for low-energy critical points of the unconstrained functional whose mass is smaller than a threshold $\\mu_0$; it is proved by combining the growth bound (f1) with the Gagliardo–Nirenberg inequality (2.1) and, in the Ambrosetti–Rabinowitz case, an energy–norm estimate. This lemma is what distinguishes genuine normalized solutions from limits that have lost mass. For multiplicity, the same penalized functional is fed into the Fountain theorem on the subspaces $Y_j$ and $Z_j$, producing levels $c_{r,j}$ that separate at large enough eigenvalues.","core_discovery":"The central claim is that the constrained problem can be solved by a two-step limit. For $r>1$, define $E_{r,\\mu}(u)=\\frac12\\int_\\Omega|\\nabla u|^2\\,dx-\\int_\\Omega F(u)\\,dx-H_{r,\\mu}(u)$ on $U_\\mu=\\{u\\in H^1_0(\\Omega):\\int_\\Omega|u|^2\\,dx<\\mu\\}$, where $H_{r,\\mu}(u)=f_r(\\int_\\Omega|u|^2\\,dx/\\mu)$ with $f_r(s)=s^r/(1-s)$; the barrier $H$ blows up as the mass approaches $\\mu$, and after a smooth truncation the functional extends to all of $H^1_0(\\Omega)$ with value $-1$ outside $U_\\mu$. Mountain-pass arguments produce Cerami sequences whose levels $c_r$ increase with $r$ and stay below $\\mu\\lambda_{1,0,0}/2$, and the limit as $r\\to\\infty$ is a critical point of the original unconstrained functional with mass at most $\\mu$. The key lemma is that under (f1)–(f3) or (f1)+(f4) there is a mass threshold $\\mu_0$ below which no low-energy critical point of the unconstrained problem exists; this rules out loss of mass and forces the limiting solution to sit exactly on the sphere $\\int_\\Omega|u|^2\\,dx=\\mu$, with Lagrange multiplier $\\lambda\\in[0,\\lambda_{1,0,0}]$. The same scheme, with Sobolev-trace inequalities replacing the usual Gagliardo–Nirenberg estimate, yields the Robin and Neumann theorems, with the caveat that constant functions solve the Neumann problem automatically.","pith_inferences":["Editorial inference: because the proof never uses Pohozaev-type identities except in the ground-state section, the penalized-mass mechanism should transplant to systems with prescribed sums of masses and to sign-changing or non-autonomous nonlinearities, as long as the small-mass nonexistence lemma can be adapted.","Editorial inference: the explicit threshold under (f1)+(f4) suggests that a quantitative version of the (f1)–(f3) threshold should be derivable from the constants $K_2$, $K_p$, $p$, and $\\lambda_{1,0,0}$ alone; if it is, the smallness condition becomes checkable rather than existential.","Editorial inference: the Neumann theorem leaves open whether the new solutions differ from constants; a Morse-index computation at small $\\mu$, or a stability analysis of the constant states, would be the natural way to decide this."],"forward_implications":["For any fixed smooth bounded domain and any continuous $f$ in the stated classes, there is a threshold $\\mu_0$ such that all masses below it are admissible; no differentiability, monotonicity trick, or blow-up analysis is needed.","On a domain rescaled to have small first eigenvalue, a fixed prescribed mass $\\mu$ becomes admissible: the corollary gives existence once $\\lambda_{1,0,0}$ is small enough, so each fixed $\\mu$ is solved by taking $\\Omega$ large.","Odd $f$ produces arbitrarily many distinct normalized solutions for sufficiently small $\\mu$ for the Dirichlet and Robin problems; for the Neumann problem the count is $m-1$ after excluding the two constant solutions.","The same penalized-functional scheme is claimed to cover Trudinger–Moser critical exponential growth in $\\mathbb{R}^2$, magnetic Schrödinger operators, biharmonic problems, and Choquard equations.","On a star-shaped domain, with (f1),(f2),(f4) and $q>2+4/N$, the solution can be chosen to be a ground state and is a local minimizer on the mass sphere."],"supporting_citations":[{"why":"Supplies the perturbation method and the mountain-pass theorem on open subsets used to obtain Cerami sequences for the penalized functional.","marker":"[15]"},{"why":"Companion source of the perturbation trick and the truncation argument used in the appendix.","marker":"[22]"},{"why":"Provides the Gagliardo–Nirenberg inequality (2.1) used in the nonexistence lemma and in the mountain-pass geometry.","marker":"[25]"},{"why":"The standard L2-supercritical normalization technique that the paper replaces for bounded domains; its mountain-pass idea is the benchmark.","marker":"[28]"},{"why":"Establishes small-mu existence for power nonlinearities via Morse theory and blow-up analysis, the approach the paper avoids for continuous f.","marker":"[40]"},{"why":"Previously proved Neumann normalized solutions under differentiability assumptions and is the comparison point for the new multiplicity claim.","marker":"[19]"},{"why":"Gives the Fountain theorem and minimax machinery used for the multiplicity statements.","marker":"[46]"},{"why":"Provides the equivalent norm and compact trace embeddings for H1(Omega) needed for the Robin and Neumann problems.","marker":"[36]"}],"fun_headline_variants":["Small L2 mass guarantees elliptic solutions on bounded domains","Tiny norm, infinite solutions: elliptic PDEs for odd f","Mass threshold unlocks normalized elliptic solutions","Bounded domains: small mass yields existence and multiplicity","Perturbation method proves many normalized solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 2.5's uniform nonexistence of low-energy unconstrained critical points with mass below $\\mu_0$; if such a critical point with mass $\\nu<\\mu_0$ existed, the limiting solution from the penalized family could converge to it, and the exact prescribed mass $\\mu$ would never be attained.","fun_headline_variants_meta":{"raw":{"variants":["Small L2 mass guarantees elliptic solutions on bounded domains","Tiny norm, infinite solutions: elliptic PDEs for odd f","Mass threshold unlocks normalized elliptic solutions","Bounded domains: small mass yields existence and multiplicity","Perturbation method proves many normalized solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000331,"raw_usage":{"total_tokens":1993,"prompt_tokens":1243,"completion_tokens":750,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":859,"completion_tokens_details":{"reasoning_tokens":676}},"tokens_in":859,"tokens_out":750,"duration_ms":9207,"temperature":1.0,"reasoning_tokens":676,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:45:12.195590+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To settle whether the central claim is right, look for a solution of $-\\Delta u=f(u)$ in $H^1_0(\\Omega)$ with $E(u)\\le\\lambda_{1,0,0}/2$ and $\\int_\\Omega u^2\\,dx=\\nu$ for some $\\nu<\\mu_0$: a single such critical point under (f1)–(f3) contradicts Lemma 2.5. Concretely, for $f(t)=|t|^{p-2}t$ with $2+4/N<p<2^*$ on the unit ball, one can check numerically whether branches of small-mass, low-energy solutions exist for arbitrarily small $\\nu$; the theorem predicts they do not.","supporting_citations":[{"cited_title":"Buffoni, M.J","cited_arxiv_id":null,"evidence_quote":"Supplies the perturbation method and the mountain-pass theorem on open subsets used to obtain Cerami sequences for the penalized functional."},{"cited_title":"Esteban, E","cited_arxiv_id":null,"evidence_quote":"Companion source of the perturbation trick and the truncation argument used in the appendix."},{"cited_title":"Fiorenza, M.R","cited_arxiv_id":null,"evidence_quote":"Provides the Gagliardo–Nirenberg inequality (2.1) used in the nonexistence lemma and in the mountain-pass geometry."},{"cited_title":"Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations , Nonlinear Anal","cited_arxiv_id":null,"evidence_quote":"The standard L2-supercritical normalization technique that the paper replaces for bounded domains; its mountain-pass idea is the benchmark."},{"cited_title":"Pierotti and G","cited_arxiv_id":null,"evidence_quote":"Establishes small-mu existence for power nonlinearities via Morse theory and blow-up analysis, the approach the paper avoids for continuous f."},{"cited_title":"Willem, Minimax Theorems, Birkh¨ auser, Boston, 1996","cited_arxiv_id":null,"evidence_quote":"Gives the Fountain theorem and minimax machinery used for the multiplicity statements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the equivalent norm and compact trace embeddings for H1(Omega) needed for the Robin and Neumann problems."}],"review_version":1}