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REVIEW 3 major objections 6 minor 41 references

Multiplicity result for mixed local and nonlocal Kirchhoff problem involving critical growth

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Pith's one-line read For the mixed operator −Δ + (−Δ)^s with a sign-changing concave term, two distinct nonnegative solutions exist in low dimensions when N+4s<6 and the parameters are small.

desk verdict New combination but two load-bearing gaps: the Nehari auxiliary map omits the Kirchhoff terms, and the second-solution energy estimate assumes f>0 near the concentration point. read the letter →

arxiv 2411.17169 v3 pith:BXRJF4NR submitted 2024-11-26 math.AP

V. M. Tripathi This is my paper · ORCID
classification math.AP MSC 35A0135A1535B3335R11
keywords MixedlocalandnonlocaloperatorsKirchhofftypeproblemCriticalnonlinearityNeharimanifoldsign-changingweightmultiplicityofsolutionsconcave-convexfractionalLaplacian
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 tackles a boundary-value problem built from the mixed operator $-\Delta + (-\Delta)^s$ — the usual Laplace operator plus a fractional one — with a coefficient $a + b\rho(u)^{2\theta-2}$ in front that depends on the norm of the solution, in the spirit of Kirchhoff's vibrating-string model. The nonlinearity is the sum of a term at critical Sobolev growth, $|u|^{2^*-2}u$, and a lower-order concave term $\lambda f(x)|u|^{p-2}u$ with $1 < p < 2$, where the weight $f$ is allowed to change sign. The paper claims two existence results: one nontrivial nonnegative solution for small $\lambda$ in every dimension $N \ge 3$, and — under the restriction $N+4s<6$, which confines the result to dimensions 3, 4, and 5 with suitably small $s$ — a second, distinct nonnegative solution for small $\lambda$ and small $b$. Why it matters: the mixed operator's best Sobolev constant is known not to be attained, so the usual compactness machinery fails at the critical level, and the paper shows the two-solution concave–convex picture still survives.

What carries the argument

The argument turns on three objects. First, the mixed Sobolev space $X^{1,2}(\Omega)$ with norm $\rho(u) = (\int_{\mathbb{R}^N}|\nabla u|^2\,dx + [u]_s^2)^{1/2}$, whose sharp embedding constant $S_{N,s}(\Omega)$ coincides with the classical $S_N$ and is not attained — a fact the paper imports, and the reason every energy estimate must land strictly below a critical threshold to recover compactness. Second, the Nehari manifold $\mathcal{N}_\lambda$, with its fibering-map decomposition into components $\mathcal{N}^+_\lambda$, $\mathcal{N}^-_\lambda$, $\mathcal{N}^0_\lambda$ corresponding to local minima, local maxima, and inflection points of the map $t \mapsto J_\lambda(tu)$; the extremal value $\lambda^*$, a generalized Rayleigh quotient (Definition 3.2), marks the range $\lambda < \lambda^*$ in which $\mathcal{N}^0_\lambda$ is empty and the two components stay separated. Third, the concentrating test functions $u_{\varepsilon,\eta} = \eta u_\varepsilon / \|\eta u_\varepsilon\|_{L^{2^*}(\Omega)}$, built from the canonical bubbles of the critical Sobolev embedding and cut off near the origin; the expansion of $J_\lambda(u_0 + r u_{\varepsilon,\eta})$ along the crossing path yields the sub-threshold bound that drives the proof of the second solution.

What would settle it

A direct sign check settles the scope of the proof: take $N=3$, $s=1/4$ (so $N+4s=4<6$), $p=3/2$, and $f \equiv -1$ in the unit ball with $f \equiv +1$ outside, which is an admissible sign-changing weight. On the bubble's support the quantity $(u_0 + r u_\varepsilon)^p - u_0^p - p u_0^{p-1} r u_\varepsilon$ is nonnegative for $p \in (1,2)$, so $-\tfrac{\lambda}{p}\int f(\cdots)$ flips sign when $f<0$ there; computing the resulting term of order $\lambda \varepsilon^{N-(N-2)p/2}$ and comparing it with the leading negative term $-C\varepsilon^{(N-2)/2}$ in equation (5.6) shows whether the sub-threshold bound $J_\lambda(u_0 + r u_{\varepsilon,\eta}) < c_\lambda$ survives. If it fails, Theorem 1.2 as stated does not follow from the given estimates for weights that are negative at the origin.

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Extended reading notes

Core claim

The central discovery is that the mixed operator $-\Delta + (-\Delta)^s$ inherits the full two-solution structure of the concave–convex problem despite its degenerate variational geometry. The author shows that the Nehari manifold $\mathcal{N}_\lambda$ splits into two nonempty components $\mathcal{N}^+_\lambda$ and $\mathcal{N}^-_\lambda$ for every $\lambda$ below an extremal value $\lambda^*$; the inflection component $\mathcal{N}^0_\lambda$ is empty there, so minimizers on each component are genuine weak solutions. The first solution is the negative-energy minimizer on $\mathcal{N}^+_\lambda$; the second is the minimizer on $\mathcal{N}^-_\lambda$, whose energy lies below the compactness threshold $c_\lambda$ of Proposition 3.7. To cross from the first solution's component into $\mathcal{N}^-_\lambda$ the author uses a continuous path $u_0 + t\, l\, u_{\varepsilon,\eta}$ built from the canonical concentrating bubbles of the critical Sobolev problem, and the energy expansion along this path (with $b$ taken of order $\varepsilon^q$, $q > N-2$) shows the infimum on $\mathcal{N}^-_\lambda$ falls below $c_\lambda$ exactly when $N+4s<6$. Since $\mathcal{N}^+_\lambda$ and $\mathcal{N}^-_\lambda$ are disjoint and closed, the two minimizers are distinct, and both are shown to be nonnegative by testing the weak formulation against negative parts.

Load-bearing premise

The proof that the energy can be pushed below the compactness threshold assumes the sign-changing weight $f$ is positive on the support of the concentrating test function, but the theorem's hypothesis only places $f$ in a Lebesgue space and allows it to change sign without any local-positivity condition near the concentration point.

Editorial extensions

If this is right

  • In dimensions $N = 3, 4, 5$ with $N+4s<6$, the problem $(P_\lambda)$ has two distinct nontrivial nonnegative weak solutions for every $\lambda \in (0, \Lambda_{00})$ and all sufficiently small $b$.
  • The first solution exists in every dimension $N \ge 3$ for $\lambda \in (0, \Lambda_0)$; the restriction $N+4s<6$ enters only through the second solution's energy estimate.
  • The second solution sits on the local-maximum branch $\mathcal{N}^-_\lambda$ of the Nehari manifold while the first sits on the local-minimum branch $\mathcal{N}^+_\lambda$, so the two solutions differ in variational character and cannot coincide.
  • Both solutions are nonnegative, obtained by testing the weak form against negative parts of the solution, even though the fractional kinetic term does not commute with taking absolute values.
  • For $\lambda \ge \lambda^*$ the Nehari set ceases to be a manifold ($\mathcal{N}^0_\lambda$ becomes nonempty), so the multiplicity statement is restricted to the sub-extremal parameter range.

Reading between the lines

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

  • Editorial: the invoked positivity of $f$ on the bubble's support is not part of Theorem 1.2's hypotheses; recentering the bubble at a point where the positive part $f^+$ has positive density would extend the argument to any sign-changing $f$, but the paper includes no such translation step.
  • Editorial: the condition $N+4s<6$ is exactly the requirement $\min\{N-2, 2-2s\} > (N-2)/2$, i.e., that the positive error terms in the bubble expansion decay faster than the negative correction; in dimension 6 and above that comparison fails, so the dimension bound is structural rather than a technical afterthought.
  • Editorial: Remark 4 suggests replacing the critical exponent by $q \in (2\theta, 2^*)$; with compact embedding replacing the threshold analysis, the same two-branch machinery should give multiplicity for all $\lambda \in (0, \Lambda^* + \varepsilon)$ and in every dimension, which is a testable extension of the paper's own framework.
  • Editorial: the same decomposition should carry over to other monotone Kirchhoff coefficients $M(t)$ with comparable growth, since the argument only uses the monotonicity of the map $h(t)$ in Proposition 3.1 and the shape of the extremal value $\lambda^*$.
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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

3 major / 6 minor

Summary. The paper studies a mixed local/nonlocal Dirichlet problem (P_lambda) with Kirchhoff coefficient M(t)=a+b t^{theta-1}, a sign-changing concave term lambda f|u|^{p-2}u, and a critical Sobolev term. The main results claim one nontrivial nonnegative solution for small lambda (Theorem 1.1) and, under N+4s<6 and b sufficiently small, two distinct nonnegative solutions (Theorem 1.2). The method is Nehari-manifold minimization with fibre maps, an extremal parameter lambda_*, a compactness threshold c_lambda, and energy estimates based on the sharp mixed Sobolev constant from [11]. The paper is clearly organized, and the proposed compactness threshold is explicit.

Significance. If the proof were complete, the work would be a useful extension of Ambrosetti-Brezis-Cerami type results to mixed local/nonlocal Kirchhoff problems, and the use of the non-achieved mixed Sobolev constant [11] is a relevant technical novelty. The paper states its hypotheses and threshold parameters precisely and gives an explicit lambda-dependent compactness level in Proposition 3.7. However, the proof currently contains several load-bearing gaps: the implicit-function argument in Lemma 3.5 does not encode the Kirchhoff terms, the second-solution estimate in Proposition 5.2 uses an unstated positivity assumption on f, and the nonnegativity conclusion in Section 4 is not derived from a critical point of the functional that was minimized. These issues are localized and appear repairable, but they block acceptance in the present form.

major comments (3)
  1. [Section 3, Lemma 3.5 and Eq. (3.3)] The auxiliary function F_u(t,v) used for the implicit function theorem is t^2 rho(u-v)^2 - lambda t^p int f|u-v|^p - t^{2*} int |u-v|^{2*}, which is the Nehari equation for the semilinear problem with M(t) equivalent to 1, not for (P_lambda). For the problem at hand the Nehari condition for t(u-v) is a t^2 rho(w)^2 + b t^{2theta} rho(w)^{2theta} = lambda t^p int f|w|^p + t^{2*} int |w|^{2*}. Consequently the denominator in (3.3) should contain (2-p)a rho(u)^2 + (2theta-p)b rho(u)^{2theta} - (2*-p) int |u|^{2*}, and the numerator should include the corresponding a- and b-terms. Because Lemma 3.6 is the step that produces a Palais-Smale sequence and is used in both Theorems 1.1 and 1.2, the existence proofs are not supported as written. This is repairable by inserting the correct F_u, but it is not a purely cosmetic typo.
  2. [Section 5, Proposition 5.2, inequality (5.5)] The reduction from (5.4) to (5.5) uses 'f > 0 in the support of u_{epsilon,eta}', but this is nowhere assumed in Theorem 1.2 or Proposition 5.2: the standing assumption is only f in L^{2*/(2*-p)} and sign-changing. Since u_{epsilon,eta} is supported in a ball around 0, this is an additional pointwise positivity assumption on f near 0. The inequality J_lambda(u0 + r u_{epsilon,eta}) < c_lambda is the mechanism for obtaining the second solution, so either the theorem must be restricted to weights that are positive near the concentration point or the concentration point must be moved into the positive set of f with all estimates redone.
  3. [Section 4, proof of Theorem 1.1, nonnegativity argument] After obtaining a critical point u0 of J_lambda, the paper introduces the positive-part functional J_lambda^+ and states that 'critical points of J_lambda are also critical points of J_lambda^+'. This is false in general: if phi is supported on {u0 < 0}, the derivative of J_lambda^+ at u0 contains no contribution from (u0^+)^{p-1} or (u0^+)^{2*-1}, while the derivative of J_lambda does. The subsequent test with phi = u0^- applies to critical points of J_lambda^+, not to the minimizer of J_lambda obtained above. A separate minimization of J_lambda^+ on its Nehari set, or an equivalent replacement argument, is needed to conclude that the solution is nonnegative. This affects the advertised conclusion 'nonnegative solutions' in both theorems.
minor comments (6)
  1. [Section 3, before Proposition 3.1] The formula for the unique critical point of m_u(t) is valid only for theta > 1; when theta = 1, m_u(t) has no positive critical point and the case should be treated separately.
  2. [Section 3, Proposition 3.7] The proposition states that {u_k} is a Palais-Smale sequence, but the proof begins by assuming that {u_k} is bounded; in the applications boundedness follows from coercivity on N_lambda, so the statement should be amended accordingly.
  3. [Section 3, proof of Lemma 3.5] The ball is written as B(0, xi) although xi is the function being constructed; this is presumably B(0, epsilon).
  4. [Section 4, Lemma 4.1] The line 'Let 0 neq u in N_lambda subset N_lambda^+' is not correct, since N_lambda is not contained in N_lambda^+; the argument should start with u in N_lambda^+.
  5. [Section 5, inequality (5.3)] Inequality (5.3) is stated for p > 2, whereas p in (1,2) throughout the paper; if the inequality is intended for the critical exponent 2*, it should be relabeled so that it is not asserted for the concave exponent.
  6. [Section 5, Lemma 5.1] Lemma 5.1 is asserted without proof; since it supplies the endpoint needed to show that the path gamma(t) meets N_lambda^-, a proof or a precise adaptation of [41] to the mixed operator should be included.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central chain is built on external mixed-operator estimates and standard Nehari arguments, with the author's prior work appearing only in non-load-bearing passages.

full rationale

The derivation is a standard Nehari-manifold and fibering-map construction. All load-bearing analytic input is taken from independent external sources: the sharp mixed Sobolev constant and its identification with the local constant are imported from [11] (Biagi--Dipierro--Valdinoci--Vecchi), the compactness-threshold strategy from [21] and [14], and the test-function estimates in Section 5 from [11] and [41]. The author's own prior paper [27] appears only as a parenthetical 'see, for instance [27]' after the formula for lambda(u) and in Remark 4 as part of a speculative subcritical extension ('[27,39]'); neither occurrence supports the proofs of Theorems 1.1 or 1.2. The extremal value lambda* is defined from the fibering map and then bounded below by an explicit Lambda_1 computed from the mixed Sobolev constant, and the thresholds Lambda_0 and Lambda_00 are chosen from this bound and from the compactness level c_lambda; they are not fitted to the conclusions. I therefore find no step in which a claimed result is equivalent by construction to its inputs. This verdict is independent of two serious non-circular correctness gaps: Lemma 3.5 defines F_u(t,v) = t^2 rho(u-v)^2 - lambda t^p integral f|u-v|^p - t^{2*} integral |u-v|^{2*}, which omits the Kirchhoff coefficients a and b in the Nehari condition, and Proposition 5.2 uses 'f > 0 in the support of u_{epsilon,eta}' although the theorem only assumes f in L^{2*/(2*-p)} and sign-changing; Lemma 3.6 also omits its proof. These are errors or omissions, not circular reductions.

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

The proof is built on standard variational machinery and imported estimates; no free parameters are fitted. The main unproved inputs are the mixed Sobolev estimates from [11] and the assumption f > 0 on the test-function support, which is not among the theorem hypotheses.

assumptions (4)
  • domain assumption The sharp mixed Sobolev constant coincides with the local one: S_{N,s}(Omega) = S_N and is not attained.
    Invoked in (2.2) and throughout the compactness estimates; the paper relies on this external result from [11].
  • standard math Compact embedding X^{1,2}(Omega) into L^r(Omega) for every r in [1, 2*).
    Used in Proposition 3.7 and Lemma 3.2; standard background for the mixed operator space.
  • standard math Brezis-Lieb lemma and Ekeland variational principle are valid in this setting.
    Used in Proposition 3.7 and Lemma 3.6 to handle critical concentration and to produce minimizing Palais-Smale sequences.
  • domain assumption Asymptotic estimates for Talenti-type test functions from [11]: rho(u_{epsilon,eta})^2 = S_{N,s} + O(epsilon^{k_{s,N}}) and related integral asymptotics.
    Used in Proposition 5.2 to push the energy of a path below the compactness threshold c_lambda; these estimates are imported from [11].

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Pith. "Pith review of Multiplicity result for mixed local and nonlocal Kirchhoff problem involving critical growth." pith.science (2026). https://pith.science/paper/BXRJF4NR

@misc{pith2026241117169,
  author       = {Pith},
  title        = {Pith review of: Multiplicity result for mixed local and nonlocal Kirchhoff problem involving critical growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BXRJF4NR}},
  note         = {Machine review of arXiv:2411.17169}
}
read the original abstract

In this paper, we study the multiplicity of nonnegative solutions for mixed local and non-local problem involving critical nonlinearity with sign changing weight. Using Nehari manifold method and fibering map analysis, we have shown existence of two solutions.

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Works this paper leans on

41 extracted references · 40 canonical work pages

  1. [11]

    Biagi, S

    S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, A Brezis -Nirenberg type result for mixed local and nonlocal operators, preprint, arXiv: 2209.07502v1. 2, 3, 4, 10, 12

    arXiv

  2. [1]

    Abatangelo, M

    N. Abatangelo, M. Cozzi, An elliptic boundary value prob lem with fractional nonlinearity, SIAM J. Math. Anal. 53 (3) (2021), 3577–3601. 2

    work page 2021

  3. [2]

    Ambrosetti, H

    A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of c oncave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519–543 (1994). 2

    work page 1994

  4. [3]

    Applebaum, L´ evy Processes and Stochastic Calculus, second ed., Camb

    D. Applebaum, L´ evy Processes and Stochastic Calculus, second ed., Camb. Stud. Adv. Math., vol. 116, Cambridge University Press, Cambridge, 2009. 2

    work page 2009

  5. [4]

    Autuori, A

    G. Autuori, A. Fiscella, P. Pucci, Stationary Kirchhoff p roblems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125 (2015), 699–7 14. 2

    work page 2015

  6. [5]

    Barles, E

    G. Barles, E. Chasseigne, A. Ciomaga, C. Imbert, Lipschi tz regularity of solutions for mixed integro- differential equations, J. Differential Equations 252 (11) ( 2012), 6012–6060. 2

    work page 2012

  7. [6]

    Barrios, E

    B. Barrios, E. Colorado, A. de Pablo, U. S´ anchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), 6133–6162. 2

    work page 2012

  8. [7]

    Barrios, E

    B. Barrios, E. Colorado, R. Servadei, F. Soria, A critica l fractional equation with concave- convex power nonlinearities Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 32(4) (2015), 875–900. 2

    work page 2015

Show all 41 references
  1. [8]

    Biagi, D

    S. Biagi, D. Mugnai, E. Vecchi, A Brezis-Oswald approach for mixed local and nonlocal operators, Commun. Contemp. Math., 26(2) 28, (2024). 2

    2024

  2. [9]

    Biagi, S

    S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, A Faber-K rahn inequality for mixed local and nonlocal operators. J. Anal. Math. 150 (2023), 405–448. 2

    2023

  3. [10]

    Biagi, S

    S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, Mixed lo cal and nonlocal elliptic operators: regularity and maximum principles, Comm. Partial Differential Equations 4 7 (3) (2022), 585–629. 2

    2022

  4. [12]

    Biagi, E

    S. Biagi, E. Vecchi, On the existence of a second positiv e solution to mixed local-nonlocal concave-convex critical problems, preprint, https://arxiv.org/abs/240 3.18424 2, 3

  5. [13]

    Brezis, E

    H. Brezis, E. Lieb, A relation between pointwise conver gence of functions and convergence of functionals, Proc. Am.Math. Soc. 88(3) (1983), 486-490. 8

    1983

  6. [14]

    K. J. Brown, Y. Zhang, The Nehari manifold for a semiline ar elliptic problem with a sign changing weight function, J. Differential Equations 193(2) (2003), 481-499 . 6

    2003

  7. [15]

    Cabr´ e, Y

    X. Cabr´ e, Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincar´ e Anal. Non Li n´ eaire 31 (2014), 23–532

    2014

  8. [16]

    Caffarelli, L

    L. Caffarelli, L. Silvestre, An extension problem relat ed to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245– 1260. 2 14 VINAYAK MANI TRIPATHI

    2007

  9. [17]

    Colorado, I

    E. Colorado, I. Peral, Semilinear elliptic problems wi th mixed Dirichlet–Neumann boundary conditions, J. Funct. Anal. 199 (2003) 468–507. 2

    2003

  10. [18]

    Dipierro, E

    S. Dipierro, E. Lippi, E. Valdinoci, (Non)local logist ic equations with Neumann conditions. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 40, 1093–1166 (2023).2

    2023

  11. [19]

    Dipierro, E

    S. Dipierro, E. Valdinoci, Description of an ecologica l niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superp osition of Brownian and L´ evy processes. Physica A 575, 126052 (2021). 2

    2021

  12. [20]

    Dipierro, E

    S. Dipierro, E. Lippi, E. Valdinoci, Linear theory for a mixed operator with Neumann conditions. Asymptot. Anal. 128, 571–594 (2022). 2

    2022

  13. [21]

    J. M. do ´O , X. He, P. K. Mishra, Fractional Kirchhoff problem with crit ical indefinite nonlinearity, Math. Nachr. 292 (2019), 615–632. 2, 7

    2019

  14. [22]

    Fiscella, E

    A. Fiscella, E. Valdinoci, A critical Kirchhoff type pro blem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170. 2

    2014

  15. [23]

    Garain, J

    P. Garain, J. Kinnunen, On the regularity theory for mix ed local and nonlocal quasilinear elliptic equations, Trans. Amer. Math. Soc. 375(8) (2022), 5393–5423. 2

    2022

  16. [24]

    Garain, A

    P. Garain, A. Ukhlov, Mixed local and nonlocal Sobolev i nequalities with extremal and associated quasilinear singular elliptic problems, Nonlinear Anal. 2 23 (2022), 113022. 2

    2022

  17. [25]

    Garc ´ ıa-Azorero, I

    J. Garc ´ ıa-Azorero, I. Peral, Multiplicity of solutio ns for elliptic problems with critical exponent or with non-symmetric term, Trans. Am. Math. Soc. 323 (2) (1991), 87 7–895. 2

    1991

  18. [26]

    Il’yasov, On extreme values of Nehari manifold metho d via nonlinear Rayleigh’s quotient, Topol

    Y. Il’yasov, On extreme values of Nehari manifold metho d via nonlinear Rayleigh’s quotient, Topol. Methods Non- linear Anal. 49 (2017), 683–714. 5

    2017

  19. [27]

    P. K. Mishra, V. M. Tripath, Nehari manifold approach fo r fractional Kirchhoff problems with extremal value of the parameter, Fract. Calc. Appl. Anal. 27 (2024), 9 19-943. 5, 13

    2024

  20. [28]

    Kirchhoff, Mechanik.Teubner, Leipzig(1883)

    G. Kirchhoff, Mechanik.Teubner, Leipzig(1883). 1

  21. [29]

    Majda, E

    A. Majda, E. Tabak, A two-dimensional model for quasige ostrophic flow: comparison with the two- dimensional Euler flow, in: Nonlinear Phenomena in Ocean Dyn amics, Los Alamos, NM, 1995, Physica D 98 (2–4) (1996) 515–522. 2

    1996

  22. [30]

    Maione, D

    A. Maione, D. Mugnai, E. Vecchi, Variational methods fo r nonpositive mixed local-nonlocal operators. Fract. Calc. Appl. Anal. 26 (2023), 943–961. 2

    2023

  23. [31]

    Nehari, On a class of nonlinear second-order differen tial equations, Trans

    Z. Nehari, On a class of nonlinear second-order differen tial equations, Trans. Amer. Math. Soc. 95 (1960), 101–123. 2

    1960

  24. [32]

    Nehari, Characteristic values associated witha cla ss of non-linear second-order differential equations, Acta Math

    Z. Nehari, Characteristic values associated witha cla ss of non-linear second-order differential equations, Acta Math. 105 (1961), 141–175. 2

    1961

  25. [33]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker ’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573. 1

    2012

  26. [34]

    Rˇ adulescu, Progress in nonlinear Kirch hoff problems [Editorial], NonlinearAnal.186 (2019), 1–5

    P.Pucci, V.D. Rˇ adulescu, Progress in nonlinear Kirch hoff problems [Editorial], NonlinearAnal.186 (2019), 1–5. 2

    2019

  27. [35]

    Pucci, M

    P. Pucci, M. Xiang, B. Zhang, Existence and multiplicit y of entire solutions for fractional p− Kirchhoff equations, Adv, Nonlinear Anal. 5 (2016), 27–55. 2

    2016

  28. [36]

    Servadei, E

    R. Servadei, E. Valdinoci, The Brezis–Nirenberg resul t for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102. 2

    2015

  29. [37]

    Servadei and E

    R. Servadei and E. Valdinoci, Fractional Laplacian equ ations with critical Sobolev exponent, Rev. Mat. Complut. 28 (2015), 655–676. 2

    2015

  30. [38]

    da Silva, A

    J. da Silva, A. Fiscella, V. Viloria, Mixed local-nonlo cal quasilinear problems with critical nonlinearities, J. Differ. Equ. 408 (2024), 494–536. 2

    2024

  31. [39]

    Silva, A

    K. Silva, A. Macedo, Local minimizer over the Nehari man ifold for a class of concave-convex problems with sing changing nonlinearity, J. Differential Equations 265 ( 2018), 1894–1921. 13

    2018

  32. [40]

    Valdinoci, From the long jump random walk to the fract ional Laplacian, Bol

    E. Valdinoci, From the long jump random walk to the fract ional Laplacian, Bol. Soc. Esp. Mat. Apl. Se MA 49 (2009,) 33–44. 2

    2009

  33. [41]

    Wu, On semilinear elliptic equations involving c ritical Sobolev exponents and sign-changing weight function, Commun

    T.-F. Wu, On semilinear elliptic equations involving c ritical Sobolev exponents and sign-changing weight function, Commun. Pure Appl. Anal. 7 (2008), 383–405. 2, 10 (Vinayak Mani Tripathi) Department of Mathematics Indian Institute of Technology Bhilai 491002, Durg, Chhattisg...

    2008

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