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Some hemivariational inequalities in the Euclidean space

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 1908.06038 v1 pith:7DJDDQ7P submitted 2019-08-16 math.AP

Giovanni Molica Bisci , Dušan D. Repovš This is my paper · ORCID
classification math.AP MSC 35A1535J6035J6535J9135A0145A0535P30
keywords hemivariationalinequalitieslocallyLipschitzfunctionalsgeneralizeddirectionalderivativesymmetriccriticalityradialandnon-radialsolutionssign-changingEuclideanspaceR^dmultiplicity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Proposition 8, Eq. (2.18)] 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.
  2. [Section 3, proof of Case 2 after (3.16)] 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.
minor comments (5)
  1. [Section 3, Eq. (3.1)] The phrase 'where kappa1 = and' is incomplete; it should read 'where kappa1 is the constant appearing in (1.1)'.
  2. [Section 3, Eq. (3.22)] 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).
  3. [Lemma 6] 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.
  4. [Section 3, Part 2] 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.
  5. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all load-bearing ingredients are external general theorems or direct estimates; the flagged gap in Proposition 8 is a proof defect, not a self-referential reduction.

full rationale

The derivation chain is self-contained in the sense required by the circularity review: no theorem is obtained by renaming its own hypotheses, no fitted parameter is relabeled as a prediction, and no load-bearing conclusion is imported solely from the present authors' prior work. The abstract critical-point tool is Theorem 3, quoted from Bonanno and Molica Bisci [11], but it is a general parameter-free nonsmooth variant of Ricceri's variational principle; the authors themselves attribute the original principle to Ricceri [31]. Even though [11] includes one of the present authors, the result is a published external mathematical statement whose assumptions do not include the target hemivariational inequality, so it functions as independent support rather than circularity. The symmetric-criticality step is taken from Krawcewicz and Marzantowicz [19], and the symmetry-subspace embedding results are quoted from Bartsch-Willem [8] and Kristaly-Morosanu-O'Regan [22], none of which are authored by the present paper's authors in a way that makes the target conclusion an input. The central bridge from critical points to weak solutions is Proposition 8's estimate (2.11). That estimate is not an identity with the definition of the problem; it is a Fatou-type bound relating the generalized derivative of the composed functional to the integral of the pointwise generalized derivative, and it is applied with W ≥ 0. The review note correctly observes that the proof of (2.18) contains an unjustified passage from limsup to a claimed pointwise limit. However, this is a mathematical gap in a proof, not circularity: the inequality is not assumed as the goal, and the gap does not consist in fitting a parameter or importing the conclusion via a self-citation. A direct limsup bound would repair the argument, as the reviewer notes. No self-definitional, fitted-input, uniqueness-imported, or ansatz-smuggling pattern is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends only on standard functional analysis tools, published critical point theorems, and the specific structural assumptions on F and W. No free parameters are fitted to data, and no new entities are introduced. The proof gap in Proposition 8 concerns the justification of inequality (2.11), which is an internal step, not an extra axiom.

assumptions (4)
  • standard math Clarke subdifferential calculus and local Lipschitz continuity of the functional, including the properties in Lemma 2.
    Used throughout Section 2 to define the generalized gradient and critical points; standard background from [13,15,17,27].
  • standard math Nonsmooth Palais principle of symmetric criticality (Theorem 4 of the paper, from Krawcewicz and Marzantowicz [19]).
    Converts critical points found on fixed-point subspaces into critical points of the full functional, a key step in both parts of Theorem 1.
  • standard math Nonsmooth Ricceri variational principle (Theorem 3, from Bonanno and Molica Bisci [11]).
    Produces critical points of the restricted functional on sublevels, used to start the proof of Theorem 1.
  • domain assumption Compact embeddings of the fixed-point subspaces Fix_{H_{d,eta_i}}(H^1(R^d)) into L^q and the intersection properties (2.3)-(2.5), quoted from Bartsch-Willem [8] and Kristaly, Morosanu, and O'Regan [22].
    These embeddings and intersection properties are necessary for weak continuity of Psi and for the mutual disjointness of the solution spaces in the multiplicity argument.

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Pith. "Pith review of Some hemivariational inequalities in the Euclidean space." pith.science (2026). https://pith.science/paper/7DJDDQ7P

@misc{pith2026190806038,
  author       = {Pith},
  title        = {Pith review of: Some hemivariational inequalities in the Euclidean space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DJDDQ7P}},
  note         = {Machine review of arXiv:1908.06038}
}
abstract

The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space $\mathbb{R}^d$ ($d\geq 3$). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group $O(d)$ and their actions on the Sobolev space $H^1(\mathbb{R}^d)$. Moreover, under an additional hypotheses on the dimension $d$ and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schr\"{o}dinger equations a concrete and meaningful example of an application is presented.

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