{"id":"8316af68-17e5-4133-bd53-9f3f5b6fb6cc","arxiv_id":"1908.06038","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a class of hemivariational inequalities in R^d with symmetric nonlinearity, small values of the parameter lambda yield multiple pairs of non-trivial, sign-changing weak solutions.","lead":"This paper proves existence of non-trivial weak solutions for a class of nonsmooth variational inequalities in Euclidean space, under growth and symmetry conditions on the nonlinearity. The result extends known methods from smooth Schrödinger equations to hemivariational inequalities, a framework used in mechanics and non-smooth analysis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 8's proof of the key estimate (2.11) is incomplete: equation (2.18) passes from a limsup to an unproved pointwise limit. The estimate is repairable by a direct limsup bound, so the correct outcome is a conditional acceptance with a requested proof fix.","rationale":"The main theorem reduces to finding critical points of Jλ on invariant subspaces and then using the nonsmooth symmetric criticality principle. The step that turns a critical point into a weak solution of (Sλ) is Proposition 9, which relies on Proposition 8's inequality (2.11). Thus (2.11) is genuinely load-bearing. The proof of Proposition 8 has an invalid line at (2.18), exactly as the reader noted. I checked whether the flaw is fatal: it is not, because the pointwise limsup of the finite-difference quotients is bounded above by F^0(u(x);v(x)) for a.e. x whenever w_j(x)→u(x) and t_j→0+, which is precisely the kind of statement the paper should have used. With that replacement, (2.11) is a standard Clarke-type inequality for integral functionals and the argument goes through. I therefore keep the CONDITIONAL verdict: the manuscript should be revised to repair (2.18) (and, as a minor point, to correct the embedding constants in (3.22) from c_q to c_{i,q}) before the proof is fully rigorous. This is a proof repair, not a demonstrated counterexample.","tokens_in":18661,"tokens_out":27502,"duration_ms":257337,"concrete_test":"Rewrite the last display in the proof of Proposition 8 exactly as Jα = ∫ W limsup_j α_j dx ≤ ∫ W F^0(u;v) dx, justifying pointwise that limsup_j [F(w_j+t_jv)−F(w_j)]/t_j ≤ F^0(u;v) for a.e. x, and verifying integrability of W F^0(u;v) from (1.1). If this substitution makes the proof of (2.11) complete, the central claim stands and the manuscript only needs a proof repair; if the pointwise inequality fails for some locally Lipschitz F satisfying (1.1)–(1.2), then the bridge from critical points to solutions of (Sλ) is broken and the main theorem is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bridge from critical points to weak solutions is the inequality Ψ_E^0(u;v) ≤ ∫ W F^0(u(x);v(x)) dx in Proposition 8, applied to −Ψ in Proposition 9. In the proof, after defining α_j(x) = [F(w_j(x)+t_j v(x))−F(w_j(x))]/t_j, the paper claims in (2.18) that Jα = ∫ W limsup_j α_j dx ≤ ∫ W lim_j α_j dx ≤ ∫ W F^0(u;v) dx. The first inequality assumes the pointwise limit of the difference quotients exists; the second assumes that limit is bounded by F^0. Neither convergence is established, so the argument as written is invalid. However, the desired conclusion Jα ≤ ∫ W F^0(u;v) follows directly from the definition of F^0: for a.e. x, w_j(x)→u(x) and t_j→0+, so limsup_j α_j(x) ≤ F^0(u(x);v(x)), and W ≥ 0 allows monotonicity of the integral. Thus the flaw is real but localized and repairable; the theorem is not shown false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves existence and multiplicity results for the hemivariational inequality (S_lambda) in R^d, d >= 3, with a locally Lipschitz nonlinearity F, F(0)=0, satisfying the growth condition (1.1) and the asymptotic condition (1.2), and a nonnegative radial weight W in L^infinity cap L^1. Theorem 1 asserts that for all sufficiently small lambda > 0 there is a nontrivial radial weak solution u_lambda such that u_lambda(x) tends to 0 as |x| tends to infinity; if F is even and d > 3, there are at least zeta_S(d) = 1 + (-1)^d + floor((d-3)/2) pairs of nontrivial solutions with distinct symmetries, of which tau_d = zeta_S(d)-1 are sign-changing when d != 5. The proof uses a nonsmooth version of the Ricceri variational principle on subspaces of H^1 invariant under subgroups of O(d), the Krawcewicz-Marzantowicz symmetric criticality principle, and group-theoretic constructions from Bartsch-Willem and Kristaly-Morosanu-O'Regan. A concrete application to nonlinear Schrodinger equations is given in Section 4.","tokens_in":1586,"tokens_out":2218,"duration_ms":200989,"significance":"If the main theorem is correct, it extends multiplicity results for scalar field equations to nonsmooth hemivariational inequalities and gives an explicit dimension-dependent count of symmetry-distinct and sign-changing solutions; this is a useful contribution to the variational theory of nonsmooth problems. The paper is well organized and the abstract framework is standard. Its most valuable features are the explicit construction of invariant subspaces and the demonstration that the nonsmooth machinery applies. However, the central estimate (2.11) in Proposition 8, which bridges critical points and weak solutions in Proposition 9, is not proved correctly as written; the gap is localized and readily repairable. The multiplicity part also contains a small but load-bearing typo in the choice of epsilon. I therefore regard the result as likely correct but needing a corrected proof before publication.","major_comments":[{"comment":"The inequality J_alpha = integral W(x) limsup_j alpha_j(x) dx <= integral W(x) lim_j alpha_j(x) dx <= integral W(x) F^0(u(x);v(x)) dx is not justified. The proof has not shown that the pointwise limit lim_j alpha_j(x) exists, and without that existence the first inequality is meaningless. This is load-bearing because Proposition 9 applies (2.11) to identify every critical point of J_lambda as a weak solution of (S_lambda). The estimate is repairable: for a.e. x, w_j(x) tends to u(x) and t_j tends to 0+, so the definition of F^0 gives limsup_j alpha_j(x) <= F^0(u(x);v(x)); since W >= 0, multiplying by W and integrating preserves the inequality. I request that (2.18) be replaced by this limsup argument.","section":"Proposition 8, Eq. (2.18)"},{"comment":"The displayed admissible range for epsilon reads 0 < epsilon < (M^2 |A| + ell I_v)/I_v, but the chain of inequalities that follows requires M|A| + 2(ell - epsilon) I_v >= 0, i.e. epsilon <= ell + M|A|/(2 I_v), where I_v is the integral of |v_sigma|^2 over the annulus. As printed, the condition is not sufficient to justify the lower bound alpha M|A|/||v_sigma||^2. This looks like a typographical error, but since it occurs in the proof of nontriviality of the solution, the correct range should be stated.","section":"Section 3, proof of Case 2 after (3.16)"}],"minor_comments":[{"comment":"The phrase 'where kappa1 = and' is incomplete; it should read 'where kappa1 is the constant appearing in (1.1)'.","section":"Section 3, Eq. (3.1)"},{"comment":"In the proof of part (a2), the constants c_q in the bound for chi(gamma_i^2) should be c_{i,q}, consistently with (3.20)-(3.21).","section":"Section 3, Eq. (3.22)"},{"comment":"The statement that Psi_e is 'in fact Lipschitz continuous on L^q(R^d)' is stronger than what is proved; the proof establishes Lipschitz continuity on bounded sets, i.e. local Lipschitz continuity.","section":"Lemma 6"},{"comment":"The assertion that |z_lambda_i(x)| tends to 0 as |x| tends to infinity for the non-radial solutions is stated without proof or reference; since these functions are only known to lie in Fix_{H_d,eta_i}(H^1), a Strauss-type estimate for multi-radial functions should be supplied or cited.","section":"Section 3, Part 2"},{"comment":"There are several minor typographical items: an extra parenthesis in the definition of (S_lambda) after u(x) phi(x); missing x-arguments in F(w_j + t_j v) in (2.18); the line in Section 4 saying f belongs to L^infinity_loc(R^d) should refer to R, not R^d; and the reference list entry for [15] has an unrelated title appended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper relies on published results by the same authors and close collaborators (Theorem 3 from Bonanno-Molica Bisci, and the group-theoretic computations from Molica Bisci [26]), but these are established references, so I do not see a circularity problem. The main issue is the faulty step in Proposition 8; the authors should be asked to supply the limsup argument. There is also an unproved decay assertion for the non-radial solutions, which should be justified or cited. No grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read on 1908.06038. The main result is a dimension-dependent multiplicity theorem for hemivariational inequalities in R^d. It extends the continuous template from Molica Bisci's earlier work to the nonsmooth setting, and that extension is the real contribution. The count of solutions with distinct symmetry structures, including sign-changing ones when d≠5, is new, and the paper uses the Bartsch-Willem/Kristály-Moroșanu-O'Regan machinery carefully. The abstract framework is assembled honestly; the references to self and close collaborators are appropriate when they are the ones who proved the cited lemmas.\n\nThe one genuine soft spot is the proof of Proposition 8, equation (2.18). As the reader and stress-test both note, the proof jumps from a limsup to a limit of difference quotients without showing that limit exists. This is not cosmetic, because inequality (2.11) is the bridge that turns critical points into weak solutions in Proposition 9. But the gap is localized and repairable: from the definition of F^0, for almost every x the limsup of the difference quotients is bounded above by F^0(u(x); v(x)), and since W≥0, integrating that pointwise bound gives the desired inequality. So the lemma is true; the written proof just needs the middle step replaced with that direct bound.\n\nThere are smaller issues: equation (3.1) has a missing value after 'κ1 =', and the sign-changing claim in part (a2) is asserted more than argued, though the antisymmetry built into the H_{d,η_i} action does deliver it. These are minor.\n\nWho gets value from this: people working on nonsmooth critical point theory and on symmetric solutions of scalar field equations. I'd bring it to a reading group if the group cares about that subfield. I'd send it to a serious referee; it is not a desk reject. The correct outcome is conditional acceptance with a request to repair (2.18).","headline":"A plausible and useful extension to nonsmooth hemivariational problems, with a localized and repairable gap in the key estimate that connects critical points to weak solutions.","tokens_in":19422,"tokens_out":4578,"would_cite":true,"duration_ms":42158,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","35J60","35J65","35J91","35A01","45A05","35P30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that hemivariational inequalities on $\\mathbb{R}^d$ admit nontrivial radial weak solutions for small parameters, and that even nonlinearities in dimensions above 3 force dimension-dependent many pairs of distinct…","keywords":["hemivariational inequalities","locally Lipschitz functionals","generalized directional derivative","symmetric criticality","radial and non-radial solutions","sign-changing solutions","Euclidean space R^d","multiplicity"],"falsifier":"A direct way to test the argument is to compute both sides of (2.11) for a nonsmooth potential satisfying (1.1) and (1.2), for example $F(s)=|s|^p$ with $1<p<2$ and $p<q$, with a radial bump weight $W$ in $d=3$. If an approximating sequence $w_j\\to u$ and $t_j\\to0^+$ can be found for which the limsup in (2.18) is strictly larger than the pointwise limsup, then the integral bound could fail and Proposition 9 would no longer turn critical points into weak solutions. A reader could check this numerically or analytically for such an $F$ and see whether the inequality survives.","tokens_in":18475,"feed_emoji":"📐","tokens_out":15595,"duration_ms":124817,"temperature":0.7,"pith_summary":"This paper studies hemivariational inequalities on the whole Euclidean space $\\mathbb{R}^d$ ($d\\ge 3$): variational inequalities whose nonlinear term is nonsmooth and is expressed through the generalized directional derivative for locally Lipschitz functions. Its main theorem states that if the nonsmooth potential $F$ grows at most like $|s|^{q-1}$ with $q\\in(2,2^*)$, oscillates strongly near zero in the sense of (1.2), and $W$ is a nonnegative radially symmetric weight, then for every sufficiently small parameter $\\lambda$ the inequality $(S_\\lambda)$ admits a nontrivial radial weak solution that decays to zero at infinity. When $F$ is even and $d>3$, the same argument yields at least $\\zeta(d)_S=1+(-1)^d+\\lfloor(d-3)/2\\rfloor$ pairs of solutions with different symmetries, and, unless $d=5$, at least $\\tau_d=(-1)^d+\\lfloor(d-3)/2\\rfloor$ of those pairs are sign-changing. The result matters because it transfers a classical existence and multiplicity picture for smooth scalar field equations to a nonsmooth setting where derivative tools must be replaced by generalized directional derivatives and symmetric criticality.","feed_headline":"Dimension dictates the number of sign-changing solutions","feed_subtitle":"Small perturbations of a nonsmooth potential yield many symmetric weak solutions; how many depends on the dimension.","key_machinery":"The central machinery is the combination of a nonsmooth symmetric criticality principle and a nonsmooth variational principle for locally Lipschitz functionals. The symmetric criticality principle says that a critical point of a $G$-invariant locally Lipschitz functional restricted to the fixed-point space of an isometric compact group action is automatically a critical point on the whole space. The variational principle produces a global minimum on a sublevel of the norm part of the energy, which then satisfies the critical-point inequality. The other load-bearing object is inequality (2.11): for the weighted composite functional $\\Psi_E(u)=\\int_{\\mathbb{R}^d}W(x)F(u(x))\\,dx$ restricted to a closed subspace $E$, the generalized directional derivative $\\Psi_E^0(u;v)$ is bounded above by $\\int_{\\mathbb{R}^d}W(x)F^0(u(x);v(x))\\,dx$. This estimate is what converts an abstract critical point into a weak solution of $(S_\\lambda)$. Compact embeddings of the radial fixed-point space and of the nonradial fixed-point spaces $Fix_{H_{d,\\eta_i}}(H^1(\\mathbb{R}^d))$ into $L^q$ supply the needed compactness on the whole space.","core_discovery":"On its own terms, the paper establishes Theorem 1. For a locally Lipschitz $F:\\mathbb{R}\\to\\mathbb{R}$ with $F(0)=0$, growth $|\\zeta|\\le \\kappa_1(1+|s|^{q-1})$ for every $\\zeta\\in\\partial F(s)$ and some $q\\in(2,2^*)$, and the condition $\\limsup_{s\\to0^+}F(s)/s^2=+\\infty$ together with a matching lower bound, the problem $(S_\\lambda)$ has, for every small $\\lambda$, a nontrivial radial weak solution $u_\\lambda\\in H^1(\\mathbb{R}^d)$ with $|u_\\lambda(x)|\\to0$ as $|x|\\to\\infty$. If $d>3$ and $F$ is even, the paper constructs $\\zeta(d)_S=1+(-1)^d+\\lfloor(d-3)/2\\rfloor$ pairs $\\{\\pm u_{\\lambda,i}\\}$ of nontrivial weak solutions lying in mutually disjoint fixed-point spaces of carefully chosen subgroups of $O(d)$, so the solutions have genuinely different symmetry structures; for $d\\ne5$, at least $\\tau_d=(-1)^d+\\lfloor(d-3)/2\\rfloor$ of those pairs are sign-changing. The proof finds critical points of a locally Lipschitz energy functional on invariant subspaces and then uses a nonsmooth symmetric criticality principle, together with an estimate for the generalized directional derivative of the weighted composite functional, to turn them into weak solutions of the hemivariational inequality.","pith_inferences":["A natural extension not pursued in the paper would be to test whether the same dimension-dependent count holds when the weight $W$ is allowed to change sign; the compactness arguments would need modification, but the group-theoretic counting is independent of positivity.","The reliance on inequality (2.11) suggests that repairing or bypassing that estimate, rather than replacing the variational framework, would be the most direct route to extending the theorem to more general nonsmooth nonlinearities, including critical growth.","Because the fixed-point subspaces used here depend only on the group actions, the same counting formula likely applies to hemivariational inequalities on strip-like domains or on manifolds with the same symmetries, as the paper itself signals as future work."],"forward_implications":["For every $\\lambda\\in(0,\\lambda^\\star)$ there is a nontrivial radial weak solution, and it decays to zero at infinity, so the inequality has homoclinic-type solutions.","With an even nonlinearity and $d>3$, the number of distinct solution pairs is at least $\\zeta(d)_S=1+(-1)^d+\\lfloor(d-3)/2\\rfloor$, so the multiplicity is dimension-dependent and parity-sensitive.","When $d\\ne5$, at least $\\tau_d=(-1)^d+\\lfloor(d-3)/2\\rfloor$ of the solution pairs are sign-changing, meaning the symmetric construction does not merely produce positive or radial profiles.","In the special case where $F$ is smooth, $(S_\\lambda)$ reduces to the parametrized scalar field equation $-\\Delta u+u=\\lambda W(x)f(u)$, so Theorem 1 is a nonsmooth extension of the classical existence and multiplicity results for that equation."],"supporting_citations":[{"why":"Supplies the nonsmooth symmetric criticality principle that lifts critical points from fixed-point subspaces to the whole space.","marker":"[19]"},{"why":"Provides the group-theoretic construction of the nonradial fixed-point subspaces and the dimension-dependent intersection properties used for multiplicity.","marker":"[22]"},{"why":"Gives the continuous-setting computations and limiting arguments that the paper adapts to the nonsmooth case.","marker":"[26]"},{"why":"Provides the nonsmooth variational principle, with sublevel minima, used to obtain the critical points.","marker":"[11]"},{"why":"Supplies the compact embeddings and intersection properties of the nonradial fixed-point subspaces.","marker":"[8]"},{"why":"Gives the compact radial embedding of the radial fixed-point subspace into $L^q$ used for radial solutions and the decay estimate.","marker":"[24]"},{"why":"Is the nonsmooth calculus reference for generalized directional derivatives and generalized gradients.","marker":"[15]"},{"why":"Is used for measurability and limiting properties of generalized directional derivatives in the proof of inequality (2.11).","marker":"[27]"}],"fun_headline_variants":["Dimension fixes number of sign-changing solutions","Symmetry spawns sign-changing pairs in Euclidean space","Nonsmooth symmetric criticality yields many weak solutions","For d>3, symmetry dictates sign-changing solution count"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is inequality (2.11), which bounds the generalized directional derivative of the weighted composite functional by the integral of the pointwise generalized derivative; its proof passes from a limsup to a claimed limit for the difference quotients in (2.18), and that passage is not shown to hold for all admissible data.","fun_headline_variants_meta":{"raw":{"variants":["Dimension fixes number of sign-changing solutions","Symmetry spawns sign-changing pairs in Euclidean space","Nonsmooth symmetric criticality yields many weak solutions","For d>3, symmetry dictates sign-changing solution count"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000917,"raw_usage":{"total_tokens":3987,"prompt_tokens":1051,"completion_tokens":2936,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":2884}},"tokens_in":667,"tokens_out":2936,"duration_ms":21437,"temperature":1.0,"reasoning_tokens":2884,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:56:56.509129+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct way to test the argument is to compute both sides of (2.11) for a nonsmooth potential satisfying (1.1) and (1.2), for example $F(s)=|s|^p$ with $1<p<2$ and $p<q$, with a radial bump weight $W$ in $d=3$. If an approximating sequence $w_j\\to u$ and $t_j\\to0^+$ can be found for which the limsup in (2.18) is strictly larger than the pointwise limsup, then the integral bound could fail and Proposition 9 would no longer turn critical points into weak solutions. A reader could check this numerically or analytically for such an $F$ and see whether the inequality survives.","supporting_citations":[{"cited_title":"Krawcewicz and W","cited_arxiv_id":null,"evidence_quote":"Supplies the nonsmooth symmetric criticality principle that lifts critical points from fixed-point subspaces to the whole space."},{"cited_title":"Krist ´aly, G","cited_arxiv_id":null,"evidence_quote":"Provides the group-theoretic construction of the nonradial fixed-point subspaces and the dimension-dependent intersection properties used for multiplicity."},{"cited_title":"Molica Bisci , A group-theoretical approach for nonlinear Schr¨ odinger e quations, Ad- vances in Calculus of Variations 54 (2015), 2985–3008","cited_arxiv_id":null,"evidence_quote":"Gives the continuous-setting computations and limiting arguments that the paper adapts to the nonsmooth case."},{"cited_title":"Bonanno and G","cited_arxiv_id":null,"evidence_quote":"Provides the nonsmooth variational principle, with sublevel minima, used to obtain the critical points."},{"cited_title":"Bartsch and M","cited_arxiv_id":null,"evidence_quote":"Supplies the compact embeddings and intersection properties of the nonradial fixed-point subspaces."},{"cited_title":"Lions , Sym´ etrie et compacit´ e dans les espaces de Sobolev, J","cited_arxiv_id":null,"evidence_quote":"Gives the compact radial embedding of the radial fixed-point subspace into $L^q$ used for radial solutions and the decay estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the nonsmooth calculus reference for generalized directional derivatives and generalized gradients."},{"cited_title":"Motreanu and P.D","cited_arxiv_id":null,"evidence_quote":"Is used for measurability and limiting properties of generalized directional derivatives in the proof of inequality (2.11)."}],"review_version":1}