{"id":"952e7ca9-1a2d-44a8-8f70-cd0385613f9b","arxiv_id":"2505.20758","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a (2,q)-Laplacian Schrödinger equation with inhomogeneous nonlinearity and prescribed L2 mass, ground states exist above a sharp mass threshold in the subcritical regime, do not exist in the critical regime, and exist on a Pohozaev manifold in the supercritical regime.","lead":"This paper proves existence, non-existence, and asymptotic behavior of ground state solutions for a double-Laplacian Schrödinger equation with forced mass and an inhomogeneous power nonlinearity. It covers the three mass regimes, gives sharp thresholds in some cases, and shows infinitely many bound states in the supercritical case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Supercritical existence (Thm 1.4, Prop 5.1) relies on the unproved claim that a weak limit of a γ(c)-minimizing sequence solves (1.1); the Lagrange multiplier rule for the Pohozaev constraint V(c) is assumed, not verified, making the natural-constraint step circular.","rationale":"The reader's verdict of CONDITIONAL is well supported. The mass-subcritical part (Theorem 1.2) appears essentially sound: the compactness argument in Proposition 3.1 is valid — the key step (3.18) works because A+B = x^{N/(N+2)}+(1−x)^{N/(N+2)} > 1 for x∈(0,1) when p>b — and the critical case c=c*_1 is handled by approximation. The real soft spot is the supercritical section. Proposition 5.1's assertion that a weak limit of a γ(c)-minimizing sequence is a weak solution is unjustified, and Lemma 5.7 assumes the very multiplier rule needed to make the Pohozaev constraint natural. This is the same load-bearing concern the reader identified, so I agree with the reader's weakest_assumption. I do not see grounds to move beyond CONDITIONAL: the gap is concrete but likely repairable by verifying surjectivity of the two-constraint differential and by deriving P(u0)=0 from the Brezis-Lieb identities and strict monotonicity of γ rather than from an unproved limiting argument. The omitted proof of Lemma 5.10 (multiplicity) is secondary, since the natural-constraint issue affects even the existence of one ground state in the supercritical range.","tokens_in":30981,"tokens_out":30235,"duration_ms":268964,"concrete_test":"Re-derive Lemma 5.7 without assuming the multipliers: for a candidate minimizer u∈V(c), compute P'(u) explicitly and verify that the linear map X→R^2, v↦(2∫uv, P'(u)v), is surjective. If it is, the standard Lagrange multiplier rule justifies (5.28); then check that the bracket in (5.33) is nonzero — for p>p*_q it is strictly negative once the coefficient is computed correctly — so μ=0 and the natural-constraint claim holds. Then revisit Proposition 5.1, obtaining P(u0)=0 from the Brezis-Lieb identity (b) combined with the strict monotonicity of γ from Lemma 5.5, rather than from the unsupported assertion that u0 solves (1.1). If the multiplier rule cannot be justified, or if the bracket in (5.33) can vanish, Theorem 1.4 as stated is not proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Proposition 5.1, after extracting a weak limit u0 of a minimizing sequence for γ(c), the text asserts: 'Since un ⇀ u0 in X, then u0 is a weak solution of Eq. (1.1)'. This does not follow: a weak limit of a minimizing sequence for I|V(c) is not automatically a critical point of I|S(c), and no Palais-Smale condition or natural-constraint argument is supplied at that point. The subsequent proof uses P(u0)=0 (from Lemma 2.2) to identify u0 as a minimizer of γ(∥u0∥2^2), and Lemma 5.5 to conclude ∥u0∥2^2=c; but Lemma 5.5 requires the minimizer to satisfy (5.10) with λ<0, which again depends on the same missing multiplier step. Lemma 5.7 assumes the existence of λ, μ in (5.28) without proving that the constraint map (u ↦ (∥u∥2^2−c, P(u))) is a submersion; in particular, the paper never checks that P'(u) is not a scalar multiple of the L2-functional v↦2∫uv. Thus the chain 'minimizer of γ(c) ⇒ weak solution of (1.1)' is circular as written. This gap affects both Theorem 1.4 and Theorem 1.5. The gap is probably fixable — for p>p*_q one can show ⟨P'(u),u⟩<0 from P(u)=0, giving P'(u)≠0 — but it is load-bearing and must be supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies L2-normalized ground states for the inhomogeneous (2,q)-Laplacian Schrödinger equation −Δu − Δ_q u = λu + |u|^{p−2}u/|x|^b on R^N, under the constraint ∫|u|^2 = c. The main results are: (i) in the mass-subcritical range, a sharp threshold c_1^* separating existence from non-existence of minimizers of the global minimization problem m(c); (ii) in the mass-critical case, non-existence of critical points for c ≤ c_2^* and non-attainment of m(c); (iii) in the mass-supercritical case, existence of minimizers for the Pohozaev-constrained problem γ(c), together with a negative Lagrange multiplier and asymptotic behavior as c → 0 and c → ∞; and (iv) a multiplicity result for bound state solutions. The proofs rely on a sharp inhomogeneous Lq-Gagliardo–Nirenberg inequality, concentration-compactness arguments, and a natural-constraint (Pohozaev manifold) approach.","tokens_in":31331,"tokens_out":22670,"duration_ms":196518,"significance":"If the results are correct, the paper provides a systematic and largely sharp theory of normalized solutions for a double-phase operator with inhomogeneous nonlinearity, including a critical mass threshold that is new in this setting. The self-contained proof of the sharp inhomogeneous Gagliardo–Nirenberg inequality is a useful contribution, and the supercritical analysis via the Pohozaev constraint is a natural extension of recent techniques. However, several load-bearing steps in the proofs are not justified as written, so the significance is conditional on substantial repair.","major_comments":[{"comment":"In Proposition 3.1 the key inequality (3.18) is invalid. Since t_0 > 1 and t_n > 1, one has t_0^{−N} + t_n^{−N} > 1; because m(c) < 0, the product m(c)(t_0^{−N}+t_n^{−N}) is strictly less than m(c). The second inequality in (3.18), which replaces this product by m(c), therefore has the wrong direction, and the claimed contradiction 'Since p > b and t_0 > 1, then (3.18) is impossible' does not follow. The pre-compactness of any minimizing sequence for m(c) and the subsequent existence statements in Theorem 1.2 are not established by the given argument.","section":"§2, Lemma 2.1 and Theorem 1.1"},{"comment":"The assertion 'Since u_n ⇀ u_0 in X, then u_0 is a weak solution of Eq. (1.1)' is unjustified. A weak limit of a minimizing sequence for I restricted to V(c) is not automatically a critical point of I|S(c), and no Palais–Smale condition, Ekeland principle, or natural-constraint argument is supplied at that point. This step is load-bearing: it is used to infer P(u_0) = 0, to identify u_0 as a minimizer of γ(∥u_0∥_2^2), and then with Lemma 5.5 to conclude ∥u_0∥_2^2 = c. Since Lemma 5.5 itself assumes the minimizer is a weak solution with a negative Lagrange multiplier, the argument is circular as written.","section":"§5.1, Proposition 5.1"},{"comment":"The proof applies the Lagrange multiplier rule to the two-constraint set V(c) = {u ∈ S(c) : P(u) = 0} but never verifies that the constraint map (u ↦ (∥u∥_2^2 − c, P(u))) is a submersion. In particular, the paper does not prove that P'(u) is not a scalar multiple of the L2-functional v ↦ 2∫uv, which is necessary for the existence of multipliers λ, μ in (5.28). Without this regularity check, the conclusion μ = 0, and hence the statement that a critical point of I|V(c) is a critical point of I|S(c), is not justified. This affects Theorem 1.4 and the ground-state interpretation of γ(c)-minimizers.","section":"§5.1, Lemma 5.7"},{"comment":"The proof of Lemma 5.4 does not treat the cases N = 1, 2 stated in part (1) of the lemma. The argument 'Since 2_b^* < q_b^*, ... if p ≤ 2_b^*, then λ_c < 0' relies on the quantity 2_b^* = 2(N−b)/(N−2), which is undefined for N = 1, 2. No separate argument is given for these dimensions, yet the negativity of λ_c for N = 1, 2 is used in Proposition 5.2 and Theorem 1.5.","section":"§5.1, Lemma 5.4"}],"minor_comments":[{"comment":"The orthogonal space V_n^⊥ is defined in H_r^1(R^3), but the ambient space in this paper is X = H^1(R^N) ∩ D^{1,q}(R^N), and the dimension N need not be 3. This appears to be a typographical artifact from a previous source and should be corrected.","section":"§5.2, Lemma 5.8"},{"comment":"There are several typos and grammatical slips, for example 'fucus', 'week solutions', 'nagetive', 'r igours proof', 'Fanally', and '22' in reference [41]. The paper would benefit from a careful proofreading.","section":"Throughout"},{"comment":"The phrase 'describes sharply' in Remark 1.1 and the phrase 'the existence, non-existence, and multiplicity of L2-normalized solutions' in the introduction are slightly awkward; consider 'describes sharp results' and 'the existence, non-existence, and multiplicity of L2-normalized solutions'. These are presentation issues only.","section":"§1, Theorem 1.2"}],"recommendation":"major_revision","confidential_remarks":"The paper advertises several sharp existence thresholds and a fairly complete picture for the inhomogeneous (2,q)-Laplacian. The stated results are plausible and likely of interest to the community, but the current manuscript has multiple load-bearing gaps: a sign error in the sharp inequality proof, an invalid compactness argument for m(c), an unjustified weak-limit step in Proposition 5.1, missing regularity for the Pohozaev constraint in Lemma 5.7, and an incomplete treatment of N = 1,2 in Lemma 5.4. These are not merely cosmetic; they undermine the central existence theorems. In my view a major revision is appropriate, and the authors should be asked to supply complete proofs of these steps. If the gaps cannot be closed, the claims should be weakened accordingly. I also recommend that the authors double-check the sign convention in the Gagliardo–Nirenberg inequality, since the displayed ground-state equation (1.7) appears to have the wrong sign for the p-Laplacian term."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper does real work: it proves a sharp inhomogeneous Lq-Gagliardo-Nirenberg inequality self-contained, and gives a fairly complete picture of the mass-subcritical and mass-critical regimes for the (2,q)-Laplacian with b>0, including sharp thresholds and a non-attainment result. That part is convincing. Second, the mass-supercritical part (Theorem 1.4) is not proved as written. In Proposition 5.1, after extracting a weak limit u0 of a minimizing sequence for gamma(c), the text says 'Since un ⇀ u0 in X, then u0 is a weak solution of Eq. (1.1).' That is false as a general statement: a weak limit of a minimizing sequence on V(c) is not automatically a critical point of I|S(c). The subsequent steps lean on this to get P(u0)=0 and then to identify u0 as a minimizer of gamma(||u0||_2^2). The same missing multiplier step reappears when Lemma 5.5 is used to force ||u0||_2^2 = c. Lemma 5.7 is supposed to handle the natural-constraint question, but it assumes Lagrange multipliers exist without checking that the constraint map (u ↦ (||u||_2^2 − c, P(u))) is a submersion at the minimizer. The stress-test note is right that the fix is probably feasible—one can get <P'(u),u> < 0 from P(u)=0, and monotonicity of gamma should then work—but the argument needs to be written carefully and the regularity of V(c) established. This is load-bearing, not cosmetic.\n\nThe multiplicity section is also thinner than the rest. Lemma 5.10, the intersection lemma needed for the fountain-theorem scheme, is not proved; it is deferred to an arXiv preprint with 'we omit the details.' For a paper that claims a multiplicity theorem, that is not enough; a referee should ask for the proof or a detailed sketch. There are also minor slips (section cross-references, the orthogonal complement in Lemma 5.8 written for H^1_r(R^3) rather than X) that are easy to fix.\n\nOn the positive side, the sharp inequality is genuinely self-contained and the equality statement is useful. The subcritical analysis, including the c*_1 threshold and the approximation argument at c=c*_1, is careful. Lemma 4.1 in the mass-critical case is clean.\n\nWho is this for? Anyone working on normalized solutions for quasilinear elliptic equations. It deserves a serious referee: the topic is active, the results are plausible and mostly well-supported, and the gaps are identifiable and probably repairable. My recommendation: send it to review, but with the expectation of a substantial revision, especially Section 5.","headline":"Systematic and mostly solid treatment of normalized (2,q)-Laplacian ground states; the supercritical existence proof has a genuine gap that is probably repairable.","tokens_in":31886,"tokens_out":9651,"would_cite":true,"duration_ms":93887,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J50","35Q41","35Q55","37K45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a class of (2,q)-Laplacian Schrödinger equations with singular inhomogeneous nonlinearity, prescribed-mass ground states exist exactly on one side of a sharp mass threshold in the subcritical range.","keywords":["normalized solutions","ground states","(2,q)-Laplacian","inhomogeneous nonlinearity","Gagliardo-Nirenberg inequality","Pohozaev identity","mass critical exponent","constrained variational methods"],"falsifier":"Choose parameters satisfying 2(2-b)/N+2 < p < p*_q and solve the constrained minimization problem numerically for masses slightly below and above the computed c*_1; finding a minimizer for some c < c*_1, or failing to find one for some c ≥ c*_1, would disprove the sharp-threshold claim. Alternatively, exhibit a minimizer of γ(c) at which the transverse derivative of the Pohozaev constraint vanishes, which would show that the natural-constraint mechanism fails and that such a minimizer need not solve equation (1.1).","tokens_in":30710,"feed_emoji":"🧮","tokens_out":9737,"duration_ms":93222,"temperature":0.7,"pith_summary":"This paper establishes existence and non-existence of L2-normalized ground states for the (2,q)-Laplacian Schrödinger equation with a singular inhomogeneous nonlinearity |u|^{p-2}u/|x|^b, under a prescribed mass constraint. In the mass-subcritical range it proves a sharp threshold: depending on p relative to 2(2-b)/N+2, minimizers of the energy on the L2-sphere exist for all masses, only for masses above a critical value c*_1, or exactly for masses at or above that critical value. In the mass-critical case p = p*_q := 2(q-b)/N+q, no minimizers exist for any mass. In the mass-supercritical case, the paper turns to a local minimization problem on the Pohozaev constraint and proves existence of minimizers that solve the equation, with precise asymptotic behavior of the energy and Lagrange multiplier, and also proves the existence of infinitely many normalized bound states. The whole analysis rests on a sharp inhomogeneous Lq-Gagliardo-Nirenberg inequality with an explicit optimizer.","feed_headline":"Double-phase Schrödinger ground states hinge on a sharp mass cutoff","feed_subtitle":"Normalized ground states exist exactly at or above a critical mass.","key_machinery":"The key machinery is a sharp inhomogeneous Lq-Gagliardo-Nirenberg inequality: for 2 < p < q(N-b)/(N-q)_+, it states that ∫ |u|^p/|x|^b dx ≤ K_{N,p,q} ||∇u||$_q^{{σ_{p,q}}$} ||u||$_2^{{p-σ_{p,q}}$}, with sharp constant K_{N,p,q} = p/||Q_{p,q}||$_2^{{p-2}}$ attained by a ground state Q_{p,q} of σ_{p,q} Δ_q Q + (p-σ_{p,q})Q - |x|^{-b}|Q|^{p-2}Q = 0. This inequality controls the singular nonlinear term, produces the lower bounds that make m(c) finite, and identifies the mass-critical exponent p*_q = 2(q-b)/N+q. The second load-bearing mechanism is the Pohozaev identity P(u) = 0, with P(u) = ||∇u||$_2^{2}$ + (N(q-2)+2q)/(2q)||∇u||_q^q - (N(p-2)+2b)/(2p)∫|u|^p/|x|^b dx, which every weak solution must satisfy and which defines the constraint V(c) for the mass-supercritical local minimization γ(c). The proof that the constrained minimizer is a true critical point on S(c) is carried by the natural-constraint argument, which uses the strict positivity of p - p*_q.","core_discovery":"The central claim is that, in the subcritical regimes, the value of the prescribed mass c relative to a critical value c*_1 := inf{c>0 : m(c)<0}, where m(c) is the infimum of the energy on the L2-sphere S(c), completely determines whether a ground state exists. Concretely, if 2(2-b)/N+2 < p < p*_q, then 0 < c*_1 < +∞ and m(c) admits a minimizer if and only if c ≥ c*_1, with the borderline minimizer present; if p = 2(2-b)/N+2, minimizers exist if and only if c > c*_1; and if 2 < p < 2(2-b)/N+2, minimizers exist for every c > 0. In the mass-critical case p = p*_q, the paper proves that m(c) = 0 for 0 < c ≤ c*_2, m(c) = -∞ for c > c*_2, and that the functional has no critical points for c ≤ c*_2, so no normalized ground state exists for any c > 0. In the mass-supercritical case p > p*_q, the paper proves that for N = 1, 2 with any p > p*_q, or for N ≥ 3 with q < 2($N^{2}$-2b)/($N^{2}$-4) and p < 2(N-b)/(N-2), the Pohozaev-constrained minimization problem γ(c) admits a minimizer, which is a genuine solution with negative Lagrange multiplier, satisfying I(u_c) → +∞ and λ_c → -∞ as c → 0+, while I(u_c) → 0 as c → ∞.","pith_inferences":["The sharp Gagliardo-Nirenberg inequality with optimizer Q_{p,q} suggests that the same threshold phenomenon should hold for the Lq-normalized version of the problem mentioned in the paper, since the authors note the arguments extend with little change; a direct adaptation would yield an Lq-analogue of the critical mass c*_1.","The paper leaves implicit that the mass-critical exponent p*_q = 2(q-b)/N+q shifts the classical threshold by exactly the inhomogeneity term 2b/N; extrapolating, other double-phase models with a |x|^{-b} weight should show a similar shift in their critical exponents.","If the transversality of the Pohozaev constraint could be verified, the hypotheses of Theorem 1.4 could likely be relaxed to the full range p < q(N-b)/(N-q)_+ for small masses, matching the paper's stated conjecture; a numerical search for minimizers of γ(c) at small c in the currently excluded range would directly test that possibility.","The non-attainment at mass-critical p means the least-energy level is realized only as a limit of concentrating or diffusing sequences, which suggests that nearby subcritical problems may exhibit instability or symmetry-breaking behavior as p approaches p*_q."],"forward_implications":["In the subcritical window 2(2-b)/N+2 < p < p*_q, the mass boundary is exact: ground states exist at and above c*_1 and fail below it, making c*_1 a genuine phase transition for the constrained energy.","At the mass-critical exponent p = p*_q, the least energy is zero for small masses and unbounded below for large masses, with no critical points at all on S(c) for c ≤ c*_2, so normalized ground states do not exist in this regime.","In the mass-supercritical regime covered by Theorem 1.4, the Pohozaev-constrained minimizer is a true solution with negative Lagrange multiplier and has the asymptotic signatures I(u_c) → ∞ as c → 0+ and I(u_c) → 0 as c → ∞.","The same method produces infinitely many normalized bound states with λ_n < 0 and I(u_n) → ∞, so the constrained problem admits not only a ground state but also excited states of arbitrarily high energy.","For subcritical masses away from the threshold, minimizing sequences are precompact, which gives orbital stability of the corresponding standing waves for the associated time-dependent problem."],"supporting_citations":[{"why":"Supplies the sharp inhomogeneous Gagliardo-Nirenberg inequality and its optimizer for the b ≠ 0 nonlinearity, the foundational estimate for the sharp threshold results.","marker":"[23]"},{"why":"Provides the prior study of L2-normalized solutions for the (2,q)-Laplacian with homogeneous nonlinearity, the baseline that the present inhomogeneous results extend and improve.","marker":"[15]"},{"why":"Supplies the concentration-compactness method used to prove precompactness of minimizing sequences and hence existence of minimizers.","marker":"[31]"},{"why":"Supplies the Pohozaev-constrained local minimization strategy and the strict monotonicity argument used in the mass-supercritical case.","marker":"[8]"},{"why":"Supplies the deformation argument that produces bounded Palais-Smale sequences under the PSP condition, used for the multiplicity theorem.","marker":"[19]"},{"why":"Supplies the fountain-type min-max scheme for producing infinitely many normalized bound states.","marker":"[14]"},{"why":"Provides the standard setting for normalized solutions, the orbital-stability interpretation, and the exponential-decay technique used in the appendix.","marker":"[16]"},{"why":"Supplies the Brezis-Lieb decomposition lemma used to handle weak limits and the convergence of the nonlinear term and of norms.","marker":"[9]"}],"fun_headline_variants":["Mass threshold decides when double-phase Schrödinger ground states exist","Critical mass marks the onset of ground states in double-phase Schrödinger","Double-phase Schrödinger ground states appear only past a critical mass","Sharp mass cutoff controls double-phase Schrödinger ground state existence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the constrained set defined by the Pohozaev identity is smooth enough for the Lagrange multiplier rule to apply at a minimizer, and that every weak limit of a minimizing sequence genuinely solves the original equation rather than merely the constraint.","fun_headline_variants_meta":{"raw":{"variants":["Mass threshold decides when double-phase Schrödinger ground states exist","Critical mass marks the onset of ground states in double-phase Schrödinger","Double-phase Schrödinger ground states appear only past a critical mass","Sharp mass cutoff controls double-phase Schrödinger ground state existence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000766,"raw_usage":{"total_tokens":3429,"prompt_tokens":1010,"completion_tokens":2419,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":626,"completion_tokens_details":{"reasoning_tokens":2349}},"tokens_in":626,"tokens_out":2419,"duration_ms":16733,"temperature":1.0,"reasoning_tokens":2349,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:49:31.013354+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose parameters satisfying 2(2-b)/N+2 < p < p*_q and solve the constrained minimization problem numerically for masses slightly below and above the computed c*_1; finding a minimizer for some c < c*_1, or failing to find one for some c ≥ c*_1, would disprove the sharp-threshold claim. Alternatively, exhibit a minimizer of γ(c) at which the transverse derivative of the Pohozaev constraint vanishes, which would show that the natural-constraint mechanism fails and that such a minimizer need not solve equation (1.1).","supporting_citations":[{"cited_title":"Genoud, An inhomogeneous, L2-critical, nonlinear Schr¨ odinger equation, Z","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp inhomogeneous Gagliardo-Nirenberg inequality and its optimizer for the b ≠ 0 nonlinearity, the foundational estimate for the sharp threshold results."},{"cited_title":"Normalized solutions to a class of $(2,q)$-Laplacian equations","cited_arxiv_id":"2212.14873","evidence_quote":"Provides the prior study of L2-normalized solutions for the (2,q)-Laplacian with homogeneous nonlinearity, the baseline that the present inhomogeneous results extend and improve."},{"cited_title":"Bellazzini, L","cited_arxiv_id":null,"evidence_quote":"Supplies the Pohozaev-constrained local minimization strategy and the strict monotonicity argument used in the mass-supercritical case."},{"cited_title":"Cingolani, K","cited_arxiv_id":null,"evidence_quote":"Supplies the deformation argument that produces bounded Palais-Smale sequences under the PSP condition, used for the multiplicity theorem."},{"cited_title":"Bartsch, S","cited_arxiv_id":null,"evidence_quote":"Supplies the fountain-type min-max scheme for producing infinitely many normalized bound states."},{"cited_title":"Cazenave, Semilinear Schr¨ odinger equations,Courant Lecture Notes in Mathematics , vol","cited_arxiv_id":null,"evidence_quote":"Provides the standard setting for normalized solutions, the orbital-stability interpretation, and the exponential-decay technique used in the appendix."},{"cited_title":"Br´ ezis, E","cited_arxiv_id":null,"evidence_quote":"Supplies the Brezis-Lieb decomposition lemma used to handle weak limits and the convergence of the nonlinear term and of norms."}],"review_version":1}