{"id":"6a9165a7-1f6a-4daf-9801-3b8552d43f84","arxiv_id":"2509.07618","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the critical fractional Choquard equation with a perturbation on a bounded star-shaped domain, at least two positive normalized solutions exist when the prescribed mass lies below an explicit threshold.","lead":"The paper proves that a fractional, nonlocal Choquard equation on a bounded star-shaped domain admits at least two positive solutions with a prescribed L2 mass. These 'normalized solutions' are central in recent nonlinear analysis, and bounded-domain results were missing for this doubly nonlocal critical equation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 overclaims: it omits the N∈(2s,6s) hypothesis that Proposition 5.1 requires for α≠0; for N≥6s the asymptotic estimate (5.1) fails.","rationale":"The reader correctly identified the one-line Pohozaev identity as a structural input, but my strongest concern is different and directly checkable from the manuscript: the statement of Theorem 1.2 is broader than what Proposition 5.1 proves. The missing hypothesis N∈(2s,6s) is not cosmetic; it is exactly the regime where the positive O(ϵ^{2s}) term from the L²-normalizing scaling overwhelms the negative HLS mixed term of order ϵ^{(N−2s)/2}. Thus for N≥6s and α≠0 the mountain-pass inequality β(d)<m_d + critical value is unsupported, so the claimed second solution is unsupported. The phantom reference to 'Theorem 2.3' strengthens the impression that the hypotheses were not synchronized. This concern does not refute the whole paper: with N∈(2s,6s) added to Theorem 1.2, the proof structure is coherent. Therefore the appropriate verdict remains CONDITIONAL, with the condition being a corrected, complete statement of the hypotheses and a verification (or repair) of the asymptotics in the borderline regime.","tokens_in":23361,"tokens_out":23216,"duration_ms":254426,"concrete_test":"Set N=6, s=1 (so N>2s and N≥6s), choose α<0 and admissible p, μ, and insert the explicit bubble v_ϵ of Lemma 5.1 into Proposition 5.1. Track the two competing orders in (5.2)–(5.3): the positive term (|α|/p)(1−b^{p(δ_p−1)})||u_d+k v_ϵ||_p^p is O(k²ϵ^{2s}) because ||v_ϵ||_2² = C_s ϵ^{2s}; the negative HLS cross term is B k^{2q−1}ϵ^{(N−2s)/2} = B k^{2q−1}ϵ^{2s}. For every fixed k the latter is comparable to, not larger than, the former, so no choice of k with ϵ→0 yields (5.1). If the authors can supply an additional cancellation, the theorem may survive; otherwise Theorem 1.2 must be restricted to N∈(2s,6s) for α≠0.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The printed Theorem 1.2 states only 2<p<2*−1 plus conditions from 'Theorem 2.3' (a nonexistent theorem; presumably Theorem 1.1), and does not restrict N beyond N>2s. Yet Proposition 5.1 explicitly assumes N∈(2s,6s) for α≠0, and the proof genuinely needs that condition. In the notation of Proposition 5.1, b^s = ||u_d + k v_ϵ||_2 / sqrt(d), and for N>4s, ||v_ϵ||_2² = C_s ϵ^{2s} by Lemma 5.1(ii). Hence 1−b^{p(δ_p−1)} is of order k² ϵ^{2s}, which enters the upper bound in (5.2)–(5.3) as an unwanted positive term of order k² ϵ^{2s} (for α<0 directly; for α>0 through the estimate 1−b^{p(δ_p−1)} < ... also of order k² ϵ^{2s}). The only negative term that can cancel this in the displayed argument is the HLS mixed term k^{2q−1} B ϵ^{(N−2s)/2} from Lemma 5.2(ii), which appears with a minus sign. If N≥6s, then (N−2s)/2 ≥ 2s, so this negative term is o(ϵ^{2s}) relative to the positive O(k² ϵ^{2s}) term. For any fixed k, as ϵ→0 the positive term dominates, and (5.1), hence Proposition 5.1 and therefore Theorem 1.2 for α≠0, is not established in that regime. This is an internal inconsistency between the theorem statement and the proof, not a mere stylistic omission.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the existence and multiplicity of normalized positive solutions to a critical fractional Choquard equation with an L^2-mass constraint on a smooth bounded star-shaped domain. The proof is variational: the authors minimize an associated energy J̃_d on the Pohozaev set T (defined by a strict inequality derived from a fractional Pohozaev identity), obtaining a ground-state solution u_d (Theorem 1.1). They then construct a second solution by a uniform mountain pass argument combined with the monotonicity trick of Jeanjean, with compactness at the mountain-pass level achieved through a strict energy comparison built on a concentrated test function v_ε (Proposition 5.1, Theorem 1.2).","tokens_in":23647,"tokens_out":26082,"duration_ms":231250,"significance":"If the proofs are completed, the paper would contribute a genuinely new result: the first treatment of normalized solutions for a critical nonlocal Choquard equation on a bounded domain, extending to a nonlocal, non-scale-invariant setting the recent bounded-domain normalized-solution literature. The strategy is standard in outline but technically demanding because of the double nonlocality and the lack of scaling invariance. The paper also contains useful structural elements: the Pohozaev-set reduction, the family of functionals J̃_{d,θ}, and the test-function estimate. However, several load-bearing steps are currently asserted rather than proved, and at least one theorem statement goes beyond what the proof establishes. With careful repair, the results are plausible; as written, the manuscript is not yet a complete proof of its stated theorems.","major_comments":[{"comment":"Theorem 1.2 is stated without any restriction on N beyond N>2s, yet both Proposition 4.2 and Proposition 5.1 explicitly assume N∈(2s,6s) when α≠0. The restriction is essential: in the notation of Proposition 5.1, b^s=∥w_{ε,k}∥₂/√d, and for N>4s, Lemma 5.1(ii) gives ∥v_ε∥₂²=C_s ε^{2s}. Hence 1−b^{p(δ_p−1)} is of order k²ε^{2s}. This enters (5.2)–(5.3) as a positive term for α>0, and the only negative term that can dominate it is the HLS mixed term k^{2q−1}B ε^{(N−2s)/2} from Lemma 5.2(ii). When N≥6s, (N−2s)/2≥2s, so this negative term is o(ε^{2s}) and cannot compensate for any fixed k as ε→0. Therefore (5.1), and hence the strict inequality β(d)<m_d+... is not established for N≥6s. The theorem statement overclaims the range of validity of the proof. Also, Theorem 1.2 refers to a nonexistent 'Theorem 2.3'; the intended reference is presumably Theorem 1.1.","section":"§1, Theorem 1.2; §5, Proposition 5.1"},{"comment":"Lemma 2.3 is the structural input that places every critical point of J̃_d on S_d^+ into the Pohozaev set T, and therefore underlies the coercivity, the gap inf_T J̃_d < inf_∂T J̃_d, and both existence theorems. As written, the identity has no term involving the Lagrange multiplier λ, even though the critical point equation contains λu. For a constrained critical point, the standard fractional Pohozaev identity for (−Δ)^s u = λu + ... contains a λ∥u∥₂² contribution; its coefficient may be nonzero, and for α>0 the sign of λ is not controlled (only λ<λ_{1,s}). If the identity should include λ∥u∥₂², the implication u∈T is not automatic. The one-line citation to [32,39] is insufficient here, because those references treat fixed-frequency problems. Please prove the identity in the constrained setting or state precisely the version being imported and explain why the λ-term is absent/irrelevant","section":"§2, Lemma 2.3 and (2.4)"},{"comment":"The first displayed relation in the proof of Proposition 5.1, J̃_d(W_{ε,k}) = J̃_d(w_{ε,k}) + (α/p)(1−b^{p(δ_p−1)})∥w_{ε,k}∥_p^p, is not derived. W_{ε,k} is obtained from w_{ε,k} by the L^2-normalizing dilation b, so the kinetic term and the HLS term acquire additional b-dependent factors; these give positive contributions of order (b^{2s}−1)∥w∥²_X0, i.e., of order k²ε^{2s} when N>4s. The proof must show that these positive terms are controlled by the negative HLS term from Lemma 5.2(ii); the condition N<6s appears to be exactly what is needed, but the estimates are not provided. The subsequent statement that for k∈(k₀,k₁) one can choose R small enough is not quantified, and the dependence of the constants B_i in Lemma 5.2 on R and on the profile u_d is not addressed. This leaves the crucial strict inequality (5.1) unsupported.","section":"§5, after (5.2)"},{"comment":"In the compactness arguments, the paper passes from the constrained critical-point equations to the identity ∥w_n∥²_X0 = ∥w_n∥^{2*}_{NL}+o_n(1) by invoking the Brezis-Lieb lemma [5]. The standard Brezis-Lieb lemma applies to L^r norms, not directly to the double integral of the Riesz potential appearing in the HLS term. A Brezis-Lieb-type lemma for the Choquard term is needed; the paper itself cites such a result only later in Lemma 5.1(v) (from [14]). Without this step being justified, the strict convergence at both the ground-state level and the mountain-pass level is incomplete. Please supply the missing nonlocal version of Brezis-Lieb and verify that its hypotheses hold for sequences in S_d^+ with bounded X₀-norm.","section":"§3, proof of Theorem 1.1; §4, Proposition 4.2"}],"minor_comments":[{"comment":"The phrase 'suppose α,d,p satisfy the conditions as in Theorem 2.3' refers to a nonexistent theorem. It should read 'as in Theorem 1.1'.","section":"§1, Theorem 1.2"},{"comment":"The sentence 'Φ'(t)>0 in (t_u,∞)' should presumably be 'Φ'(t)<0 in (t_u,∞)'; otherwise it contradicts the preceding uniqueness statement and the decay of Φ(t) as t→∞.","section":"§2, Proposition 2.2(iii)"},{"comment":"The condition '2<p<2*−1' is ambiguous: it should specify whether 2* denotes 2*_s or 2*_{μ,s}, and how it interacts with the hypotheses in (A2) and (A3).","section":"§1, Theorem 1.2"},{"comment":"The notation ∥u^+∥^{2*_{μ,s}}_{NL} is used for the HLS double integral without a definition. Since this object is not an ordinary L^r norm, it should be defined explicitly, e.g., I(u)=∫∫ |u(x)|^{2*}|u(y)|^{2*}/|x−y|^μ dxdy.","section":"Throughout"},{"comment":"The line 'u_n → u_d strongly in L^p(Ω), for 2≤p<2*_{μ,s}' is slightly misleading: since 2*_{μ,s}<2*_s, compactness holds for the larger range 2≤p<2*_s. This is not a mathematical error, but the stated range should be justified or corrected.","section":"§3, proof of Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a timely and interesting problem, and the overall strategy is likely salvageable. However, the published version cannot state Theorem 1.2 in its current form, and the proof of Theorem 1.1 depends on a Pohozaev identity whose constrained version is not established. The authors should also be asked to check carefully the scaling estimates in Proposition 5.1; the N<6s restriction is not a stylistic detail but a genuine requirement of the argument. I would not recommend rejection, as the identified gaps appear fixable within the scope of the paper, but they require substantial rewriting."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the headline: this is a genuine new result in a small niche—first normalized-solution existence and multiplicity for a doubly nonlocal critical Choquard problem on a bounded star-shaped domain—and the proof architecture is mostly the standard one (Pohozaev set T, monotonicity trick, mountain pass). But the paper is not ready as posted. The main theorem for the second solution overclaims: Theorem 1.2 omits the N∈(2s,6s) hypothesis that Proposition 5.1 explicitly requires when α≠0. For N≥6s the key estimate (5.1) is not established; the positive O(k²ε^{2s}) term from the mass-scaling factor can dominate the HLS mixed term, which is only O(ε^{(N−2s)/2}). That is a missing hypothesis in the statement, not a collapse of the whole construction.\n\nWhat is genuinely good: the two-theorem result (a positive normalized solution via minimization over T, then a second mountain-pass solution) is not in the cited literature—[15] is R^N, [43] is the local s=1 template. The internal algebra in Proposition 2.2 and the Φ(t) scaling argument is coherent. The self-citations for regularity and the HLS expansion are appropriate uses, not padding.\n\nSoft spots, in rough order of seriousness:\n\n1. Lemma 2.3, the fractional Pohozaev identity with boundary term, is one line: 'we have the desired result using [32,39].' This identity is load-bearing because the sign of x·ρ puts critical points into T. It needs a precise statement or a full derivation tailored to this functional and constraint.\n\n2. In the proof of Theorem 1.1, the passage from a minimizing sequence in T to a constrained critical point is dispatched as 'trivial arguments,' and then strong convergence is asserted as part of the conclusion. That's the central fixed-point step; it should be written out.\n\n3. Typos and internal inconsistencies: Theorem 1.2 cites a nonexistent 'Theorem 2.3'; Proposition 2.2(iii) says Φ′>0 on (t_u,∞) where it should be <0; equation (2.8) has an inconsistent S_HL exponent. These are easy to fix but currently obstruct verification.\n\nNone of this is fatal; the strategy is sound and the estimates I checked are coherent. The paper needs the missing calculations and a corrected Theorem 1.2 before a referee can certify it. Send it to peer review rather than desk-rejecting; a serious referee can sort out the gaps. I would not cite it in its present form, but I'd keep an eye on the revision.","headline":"Genuinely new first results on normalized fractional Choquard in bounded domains, but Theorem 1.2 overclaims as printed — it omits the N∈(2s,6s) hypothesis the proof needs — and several load-bearing steps are left as citations or 'trivial arguments.'","tokens_in":24292,"tokens_out":12646,"would_cite":false,"duration_ms":117389,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","35J20","35J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"A mass-constrained critical fractional Choquard equation on star-shaped bounded domains has two positive solutions when the prescribed mass is small enough.","keywords":["normalized solutions","fractional Laplacian","Choquard equation","star-shaped bounded domain","Hardy–Littlewood–Sobolev critical growth","Pohozaev identity","mountain pass","prescribed L² norm"],"falsifier":"Take the unit ball, α = 0, and some d below threshold (1.4); compute the ground state u_d solving (F_d) numerically and evaluate Lemma 2.3's identity directly: the boundary integral (Γ(1+s)²/2)∫_{∂B}(∂u_d/δ^s)²(x·ρ)dσ must be strictly positive and the identity must balance to numerical precision, placing u_d in T with energy m_d below inf_{∂T} J̃_d. A violation of any of these checks would falsify the structural lemma on which both existence theorems rest.","tokens_in":23086,"feed_emoji":"🧮","tokens_out":19910,"duration_ms":185701,"temperature":0.7,"pith_summary":"This paper claims the first existence and multiplicity results for normalized (fixed L²-mass) solutions of a critical fractional Choquard equation on a bounded domain. Because the domain is bounded, the scaling invariance that drives whole-space constructions is lost; the authors compensate with a Pohozaev-type identity whose boundary term is positive on star-shaped domains, forcing every critical point into a constraint set T on which the energy is coercive. When the prescribed mass d lies below the explicit thresholds (1.4)–(1.6), a positive solution exists as the minimizer over T (Theorem 1.1), with the Lagrange multiplier positive for α ≤ 0 and below the first Dirichlet eigenvalue for α > 0. Under the additional conditions 2 < p < 2*_s − 1 and, for α ≠ 0, N ∈ (2s, 6s), a second positive solution of mountain-pass type exists (Theorem 1.2). If correct, the result shows that normalized-solution theory, previously confined to R^N, extends to bounded domains through geometry rather than scaling.","feed_headline":"Two positive solutions for normalized fractional Choquard equation","feed_subtitle":"On star-shaped bounded domains, small prescribed masses yield a ground state plus a mountain-pass solution.","key_machinery":"The load-bearing object is the Pohozaev identity of Lemma 2.3: every critical point of J̃_d on the mass sphere S_d^+ satisfies ||u||²_{X0} − (Γ(1+s)²/2)∫_{∂Ω}(∂u/δ^s)²(x·ρ)dσ = αδ_p||u^+||_p^p + ||u^+||^{2*_{μ,s}}_{NL}. Because Ω is star-shaped, x·ρ > 0 on the boundary, so the boundary integral is strictly positive and every critical point lands in the coercivity set T of (2.4) — the bounded-domain stand-in for the whole-space Pohozaev manifold. For the second solution, the machinery is Jeanjean's monotonicity trick on the homotopy J̃_{d,θ} (θ ∈ [1/2,1]), plus cut-off HLS bubbles v_ϵ = ζU_ϵ (Lemmas 5.1–5.2) that push the mountain-pass level below m_d + ((2*_{μ,s}−1)/(22*_{μ,s})) S_HL^{2*_{μ,","core_discovery":"The paper claims that the mass-constrained critical fractional Choquard problem (F_d) on a bounded star-shaped domain has a positive solution realizing m_d = inf_T J̃_d, with T the Pohozaev set of (2.4); and, when 2 < p < 2*_s − 1 (and N ∈ (2s,6s) if α ≠ 0), a second positive solution at the mountain-pass level β(d). The multiplier λ_d is positive for α ≤ 0, below the first Dirichlet eigenvalue for α > 0. The mass thresholds (1.4)–(1.6) enforce inf_T J̃_d < inf_{∂T} J̃_d, keeping minimizers off ∂T, so Ekeland's principle yields a critical point; Jeanjean's monotonicity trick supplies the bounded Palais–Smale sequence at β(d), and a strict estimate below the Hardy–Littlewood–Sobolev critical","pith_inferences":["Editorial extension: the same construction — a Pohozaev identity with positive boundary term defining a coercivity set T — should transfer to other nonlocal critical problems with prescribed mass on star-shaped domains (systems, mixed-order operators, other Riesz-kernel nonlinearities), since the paper's mechanism never uses the specific Choquard kernel except through the Hardy–Littlewood–Sobolev ","Editorial observation: the method is geometric in an essential way — if the domain is not star-shaped, the boundary integral in Lemma 2.3 can change sign, so T would no longer contain all critical points; extending the results to general bounded domains would require a different mechanism, and the paper leaves that open.","Testable direction: evaluating the suprema in (1.4)–(1.6) on simple domains such as the unit ball would yield explicit mass ranges; probing d near the threshold would show whether the gap inf_T J̃_d < inf_{∂T} J̃_d closes continuously, indicating a sharp condition, or whether the thresholds are merely sufficient."],"forward_implications":["Normalized Choquard theory, previously developed on R^N, now covers bounded star-shaped domains: the Pohozaev set T replaces the Pohozaev manifold, with star-shapedness supplying the sign that scaling supplied in the whole space.","The thresholds (1.4)–(1.6) are explicit conditions: any prescribed mass d below the displayed supremum is admissible, giving a checkable range of masses for which a ground state exists.","For α > 0 the ground state's Lagrange multiplier is strictly below the first Dirichlet eigenvalue λ_{1,s} of (−Δ)^s, while for α ≤ 0 it is positive — the sign of the local power nonlinearity determines which side of the spectrum the multiplier sits on.","The second solution coexists with the minimizer at a strictly higher mountain-pass level β(d), which remains below the compactness threshold m_d + ((2*_{μ,s}−1)/(22*_{μ,s})) S_HL^{2*_{μ,s}/(2*_{μ,s}−1)}; the theorem additionally requires 2 < p < 2*_s − 1 and, for α ≠ 0, N ∈ (2s, 6s)."],"supporting_citations":[{"why":"Provides the fractional Choquard Pohozaev identity used as Lemma 2.3, the Hardy–Littlewood–Sobolev sharp constant S_HL and its optimizer (Lemma 2.1), and the bubble estimates imported in Lemma 5.1.","marker":"[32]"},{"why":"Source of the boundary-integral form of the fractional Pohozaev identity; the positivity of its x·ρ boundary term (star-shapedness) is what defines the set T.","marker":"[39]"},{"why":"Supplies the monotonicity trick (Proposition 4.1) used to obtain a bounded Palais–Smale sequence at the mountain-pass level for the homotopy J̃_{d,θ}.","marker":"[17]"},{"why":"Provides Lemma 5.2 (pairing estimates for the truncated bubbles against arbitrary test functions) and the two-normalized-solutions-on-star-shaped-domains strategy adapted in Section 5.","marker":"[43]"},{"why":"Supplies the Gagliardo–Nirenberg inequality (Lemma 2.2) used to bound ∥u^+∥_p^p on ∂T and in the coercivity and gap estimates.","marker":"[10]"},{"why":"Provides the Hardy–Littlewood–Sobolev inequality (Proposition 2.1) and its sharp constant underlying the critical exponent 2*_{μ,s} and S_HL.","marker":"[21]"},{"why":"The whole-space normalized ground-state result for the critical fractional Choquard equation with local perturbation; its estimates are cited for Lemma 5.1(i)–(iv) and it anchors the construction being adapted to bounded domains.","marker":"[15]"},{"why":"Cited for the cut-off bubble asymptotics in Lemma 5.1(i)–(iv); those expansions place the mountain-pass level below the critical Hardy–Littlewood–Sobolev threshold in Proposition 5.1.","marker":"[40]"},{"why":"The Brézis–Lieb lemma, used in the proofs of Theorems 1.1 and 1.2 to split the energy of weakly convergent subsequences and force strong convergence.","marker":"[5]"}],"fun_headline_variants":["Critical Choquard on bounded domain admits two positive states","Ground state plus mountain-pass solution for normalized Choquard","Two normalized solutions for critical fractional Choquard equation","Star-shaped domain gives two solutions for critical Choquard","Mass-constrained fractional Choquard: multiplicity of positive states"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"Everything rests on the Pohozaev-type identity of Lemma 2.3, which the paper cites in one line from prior work: every critical point on the fixed-mass sphere satisfies an energy balance including a boundary integral, and the strict positivity of that integral — guaranteed only by star-shapedness — is what places all critical points in the set T. If the identity is misstated, or the boundary term could vanish or change sign on a smooth bounded domain, then T may be empty and b","fun_headline_variants_meta":{"raw":{"variants":["Critical Choquard on bounded domain admits two positive states","Ground state plus mountain-pass solution for normalized Choquard","Two normalized solutions for critical fractional Choquard equation","Star-shaped domain gives two solutions for critical Choquard","Mass-constrained fractional Choquard: multiplicity of positive states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000271,"raw_usage":{"total_tokens":1513,"prompt_tokens":841,"completion_tokens":672,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":605}},"tokens_in":585,"tokens_out":672,"duration_ms":7183,"temperature":1.0,"reasoning_tokens":605,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T21:59:25.948040+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the unit ball, α = 0, and some d below threshold (1.4); compute the ground state u_d solving (F_d) numerically and evaluate Lemma 2.3's identity directly: the boundary integral (Γ(1+s)²/2)∫_{∂B}(∂u_d/δ^s)²(x·ρ)dσ must be strictly positive and the identity must balance to numerical precision, placing u_d in T with energy m_d below inf_{∂T} J̃_d. A violation of any of these checks would falsify the structural lemma on which both existence theorems rest.","supporting_citations":[{"cited_title":"Fractional Choquard equation with critical nonlinearities","cited_arxiv_id":null,"evidence_quote":"Provides the fractional Choquard Pohozaev identity used as Lemma 2.3, the Hardy–Littlewood–Sobolev sharp constant S_HL and its optimizer (Lemma 2.1), and the bubble estimates imported in Lemma 5.1."},{"cited_title":"The Pohozaev identity for the fractional Laplacian","cited_arxiv_id":null,"evidence_quote":"Source of the boundary-integral form of the fractional Pohozaev identity; the positivity of its x·ρ boundary term (star-shapedness) is what defines the set T."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the monotonicity trick (Proposition 4.1) used to obtain a bounded Palais–Smale sequence at the mountain-pass level for the homotopy J̃_{d,θ}."},{"cited_title":"Uniqueness of radial solutions for the fractional Laplacian.Communications on Pure and Applied Mathematics, 69(9):1671–1726, 2016","cited_arxiv_id":null,"evidence_quote":"Supplies the Gagliardo–Nirenberg inequality (Lemma 2.2) used to bound ∥u^+∥_p^p on ∂T and in the coercivity and gap estimates."},{"cited_title":"Normalized ground states for the critical fractional Choquard equation with a local perturbation.The Journal of Geometric Analysis, 32(10):252, 2022","cited_arxiv_id":null,"evidence_quote":"The whole-space normalized ground-state result for the critical fractional Choquard equation with local perturbation; its estimates are cited for Lemma 5.1(i)–(iv) and it anchors the construction being adapted to bounded domains."},{"cited_title":"The Brezis-Nirenberg result for the fractional Laplacian.Transactions of the American Mathematical Society, 367(1):67–102, 2015","cited_arxiv_id":null,"evidence_quote":"Cited for the cut-off bubble asymptotics in Lemma 5.1(i)–(iv); those expansions place the mountain-pass level below the critical Hardy–Littlewood–Sobolev threshold in Proposition 5.1."},{"cited_title":"A relation between pointwise convergence of functions and convergence of functionals.Proceedings of the American Mathematical Society, 88(3):486– 490, 1983","cited_arxiv_id":null,"evidence_quote":"The Brézis–Lieb lemma, used in the proofs of Theorems 1.1 and 1.2 to split the energy of weakly convergent subsequences and force strong convergence."}],"review_version":1}