Eigenvalue problems with unbalanced growth: Nonlinear patterns and standing wave solutions

Du\v{s}an Repov\v{s}; Istv\'an Farag\'o

arxiv: 1906.09001 · v1 · pith:MUJOO6OGnew · submitted 2019-06-21 · 🧮 math.AP

Eigenvalue problems with unbalanced growth: Nonlinear patterns and standing wave solutions

Istv\'an Farag\'o , Du\v{s}an Repov\v{s} This is my paper

Pith reviewed 2026-05-25 18:59 UTC · model grok-4.3

classification 🧮 math.AP

keywords nonlinear eigenvalue problemsdouble-phase energyunbalanced growthvariational methodsPucci-Serrin maximum principleexistence and non-existencelack of compactnessstanding waves

0 comments

The pith

Nonlinear eigenvalue problems with double-phase energy admit or exclude solutions depending on growth parameters and eigenvalue ranges.

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

The paper studies two classes of nonlinear eigenvalue problems whose energy functionals combine two distinct growth rates, leading to a lack of compactness in the underlying function space. It establishes existence and non-existence of solutions together with related qualitative properties by combining variational methods with an extension of the Pucci-Serrin maximum principle to the double-phase setting. A reader cares because these problems arise in models of unbalanced growth, where standard compactness arguments fail and the choice of growth exponents directly controls whether solutions appear. The results clarify how the interplay between the two phases and the eigenvalue parameter governs the existence landscape.

Core claim

We establish existence and non-existence results and related properties of solutions for two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. Our analysis combines variational methods with the generalized Pucci-Serrin maximum principle.

What carries the argument

Double-phase energy functional, whose integrand switches between two different power growths, combined with the generalized Pucci-Serrin maximum principle adapted to this setting to recover compactness or bounds.

If this is right

Solutions exist for eigenvalue parameters inside certain intervals determined by the two growth exponents.
Solutions cease to exist outside those intervals for the same growth conditions.
Any solution obtained satisfies additional regularity or positivity properties derived from the adapted maximum principle.
The lack of compactness is overcome precisely when the double-phase structure and the maximum principle interact favorably.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar existence thresholds may appear in other unbalanced-growth problems that replace the double-phase integrand with a variable-exponent version.
The same variational-plus-maximum-principle strategy could be tested on systems where the two phases act on different components of the solution vector.
Numerical approximation schemes for these eigenvalue problems would need to respect the two-phase structure to avoid artificial solutions outside the proven existence ranges.

Load-bearing premise

The generalized Pucci-Serrin maximum principle continues to hold for the double-phase energy under the specific growth and regularity conditions placed on the functional.

What would settle it

A concrete double-phase integrand satisfying the paper's growth hypotheses for which a positive solution violates the maximum principle bound would falsify the existence claims.

read the original abstract

We consider two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. We establish existence and non-existence results and related properties of solutions. Our analysis combines variational methods with the generalized Pucci-Serrin maximum principle.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper applies known variational methods plus a generalized maximum principle to double-phase eigenvalue problems; the results look like straightforward extensions rather than new technology.

read the letter

The core contribution is existence and non-existence statements for two classes of nonlinear eigenvalue problems whose energy has double-phase growth and lacks compactness. The approach combines standard variational arguments with the generalized Pucci-Serrin maximum principle. That combination is the only real novelty claimed, and it is modest: the abstract presents it as an application of existing tools to a specialized setting rather than a new method or resolution of an open question. The work is honest and clearly framed, which is worth noting. The authors state the problems, the lack-of-compactness issue, and the two-pronged strategy without overclaiming. On the soft side, the abstract gives no derivation steps, no explicit hypotheses on the growth exponents or regularity that would let the maximum principle carry over, and no indication of how the lack of compactness is actually overcome. Without those details it is impossible to judge whether the argument closes or whether hidden restrictions make the results narrower than they appear. The citation pattern is not visible here, but the reliance on prior variational theory is explicit. This paper is for people already working inside double-phase variational problems who want to see one more concrete application. A reader outside that niche will not find new techniques or broader implications. It is the sort of incremental result that deserves a serious referee to check the precise hypotheses and the compactness handling, even if the final verdict is likely to be minor revision rather than rejection.

Referee Report

0 major / 1 minor

Summary. The manuscript considers two classes of nonlinear eigenvalue problems with double-phase energy functionals that exhibit lack of compactness. It claims to establish existence and non-existence results together with related properties of solutions, with the analysis combining variational methods and the generalized Pucci-Serrin maximum principle.

Significance. If the claimed extension of the generalized Pucci-Serrin principle to the double-phase setting holds under the stated growth and regularity conditions, the results would contribute to the theory of eigenvalue problems with unbalanced growth by providing existence/non-existence criteria in a setting where standard compactness arguments fail. The approach may be of interest for related problems involving standing waves and nonlinear patterns.

minor comments (1)

The abstract provides no explicit statement of the precise hypotheses on the double-phase operator, the form of the nonlinearity, or the conditions under which the generalized Pucci-Serrin principle is asserted to hold; this makes it difficult to verify applicability of the cited tools without the full text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and summary of our work on nonlinear eigenvalue problems with double-phase functionals. We note that the report expresses uncertainty in the recommendation but lists no specific major comments. We are prepared to address any concerns about the validity of the generalized Pucci-Serrin principle extension or related technical points if they are provided.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's abstract and description indicate that existence and non-existence results for the eigenvalue problems are obtained by combining standard variational methods with an extension of the Pucci-Serrin maximum principle to the double-phase setting. No equations, fitted parameters, self-definitions, or load-bearing self-citations appear in the provided text. The derivation relies on external variational theory and maximum principle results rather than reducing any claim to its own inputs by construction, so the central results remain independent of the paper's own fitted quantities or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard background results in variational methods and elliptic PDE theory; no free parameters, ad-hoc axioms, or invented entities are visible.

axioms (2)

domain assumption Variational methods apply to double-phase energy functionals despite lack of compactness
Invoked implicitly when the abstract states that existence follows from variational methods.
domain assumption Generalized Pucci-Serrin maximum principle holds in the double-phase setting
Central tool named in the abstract for obtaining solution properties.

pith-pipeline@v0.9.0 · 5556 in / 1214 out tokens · 20293 ms · 2026-05-25T18:59:51.535293+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our analysis combines variational methods with the generalized Pucci-Serrin maximum principle.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

double-phase energy functional Pp,q(u) := ∫(|∇u|^p + a(x)|∇u|^q) dx

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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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), 699-714

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A. Azzollini, Minimum action solutions for a quasilinea r equation, J. Lond. Math. Soc. 92 (2015), 583-595

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Azzollini, P

A. Azzollini, P. d’Avenia, A. Pomponio, Quasilinear ell iptic equations in RN via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Diﬀerential Equations 49 (2014), 197-213

work page 2014
[4]

Badiale, L

M. Badiale, L. Pisani, S. Rolando, Sum of weighted Lebesg ue spaces and nonlinear elliptic equations, Nonlinear Diﬀer. Equ. Appl. (NoDEA) 18 (2011), 369-405

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[5]

Bahrouni, V.D

A. Bahrouni, V.D. R˘ adulescu, D.D. Repovˇ s, Double phas e transonic ﬂow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity 32:7 (2019), 2481- 2495

work page 2019
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J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337-403. 12 I. F ARAG´O AND D. REPOV ˇS

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J.M. Ball, Discontinuous equilibrium solutions and cav itation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557-611

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P. Baroni, M. Colombo, G. Mingione, Nonautonomous funct ionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), no. 3, 347-379

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Baroni, M

P. Baroni, M. Colombo, G. Mingione, Regularity for gener al functionals with double phase, Calc. Var. (2018), 57:62

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Berestycki, P.L

H. Berestycki, P.L. Lions, Nonlinear scalar ﬁeld equat ions, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-345

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Cencelj, V.D

M. Cencelj, V.D. R˘ adulescu, D.D. Repovˇ s, Double phas e problems with variable growth, Nonlinear Anal. 177 (2018), part A, 270-287

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Chorﬁ, V.D

N. Chorﬁ, V.D. R˘ adulescu, Standing wave solutions of a quasilinear degenerate Schr¨ odinger equation with unbounded potential, Electron. J. Qual. Theory Diﬀer. Equ. 2016, Paper No. 37, 12 pp

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De Filippis, Higher integrability for constrained m inimizers of integral functionals with (p, q)-growth in low dimension, Nonlinear Anal

C. De Filippis, Higher integrability for constrained m inimizers of integral functionals with (p, q)-growth in low dimension, Nonlinear Anal. 170 (2018), 1-20

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DiBenedetto, C1+α-local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl

E. DiBenedetto, C1+α-local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl. 7 (1983), 827-850

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Filippucci, P

R. Filippucci, P. Pucci, V.D. R˘ adulescu, Existence an d non-existence results for quasilinear elliptic exterior problems with nonlinear boundary condit ions, Comm. Partial Diﬀerential Equations 33 (2008), 706-717

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Fuˇ cik, J

S. Fuˇ cik, J. Neˇ cas, J. Souˇ cek, V. Souˇ cek,Spectral Analysis of Nonlinear Operators , Lecture Notes in Mathematics, vol. 346, Springer-Verlag, Berlin-N ew York, 1973

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Marcellini, On the deﬁnition and the lower semiconti nuity of certain quasiconvex integrals, Ann

P. Marcellini, On the deﬁnition and the lower semiconti nuity of certain quasiconvex integrals, Ann. Inst. H. Poincar´ e, Anal. Non Lin´ eaire3 (1986), 391-409

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P. Marcellini, Regularity and existence of solutions o f elliptic equations with p, q–growth conditions, J. Diﬀerential Equations 90 (1991), 1-30

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Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s, Dou ble-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), no. 4, Art. 108, 21 pp

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Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s,Nonlinear Analysis – Theory and Methods , Springer Monographs in Mathematics, Springer, Cham, 2019

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Pucci, J

P. Pucci, J. Serrin, The strong maximum principle revis ited, J. Diﬀerential Equations 196 (2004), no. 1, 1-66

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Pucci, J

P. Pucci, J. Serrin, The Maximum Principle , Progress in Nonlinear Diﬀerential Equations and their Applications, Vol. 73, Birkh¨ auser, Basel, 2007

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Pucci, J

P. Pucci, J. Serrin, Maximum principles for elliptic pa rtial diﬀerential equations, Handbook of diﬀerential equations: stationary partial diﬀerential equations, Vol. IV, 355-483, Handb. Diﬀer. Equ., Elsevier/North-Holland, Amsterdam, 2007

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Pucci, R

P. Pucci, R. Servadei, On weak solutions for p-Laplacian equations with weights, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 18 (2007), no. 3, 257-267

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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

work page 2016
[26]

R˘ adulescu, D

V. R˘ adulescu, D. Repovˇ s, Partial Diﬀerential Equations with Variable Exponents: Va ria- tional Methods and Qualitative Analysis , CRC Press, Taylor & Francis Group, Boca Raton FL, 2015

work page 2015
[27]

Yu, Nonlinear p-Laplacian problems on unbounded domains, Proc

L.S. Yu, Nonlinear p-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc. 115 (1992), 1037-1045

work page 1992
[28]

Zhikov, Averaging of functionals of the calculus o f variations and elasticity theory, Izv

V.V. Zhikov, Averaging of functionals of the calculus o f variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675-710; English translation in Math. USSR- Izv. 29 (1987), no. 1, 33-66. EIGENV ALUE PROBLEMS WITH UNBALANCED GROWTH 13 (I. Farag´ o)MTA-EL TE NumNet Research Group, Budapest, Hungary & Department of Applied Analysi...

work page 1986

[1] [1]

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), 699-714

work page 2015

[2] [2]

Azzollini, Minimum action solutions for a quasilinea r equation, J

A. Azzollini, Minimum action solutions for a quasilinea r equation, J. Lond. Math. Soc. 92 (2015), 583-595

work page 2015

[3] [3]

Azzollini, P

A. Azzollini, P. d’Avenia, A. Pomponio, Quasilinear ell iptic equations in RN via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Diﬀerential Equations 49 (2014), 197-213

work page 2014

[4] [4]

Badiale, L

M. Badiale, L. Pisani, S. Rolando, Sum of weighted Lebesg ue spaces and nonlinear elliptic equations, Nonlinear Diﬀer. Equ. Appl. (NoDEA) 18 (2011), 369-405

work page 2011

[5] [5]

Bahrouni, V.D

A. Bahrouni, V.D. R˘ adulescu, D.D. Repovˇ s, Double phas e transonic ﬂow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity 32:7 (2019), 2481- 2495

work page 2019

[6] [6]

Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch

J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337-403. 12 I. F ARAG´O AND D. REPOV ˇS

work page 1976

[7] [7]

Ball, Discontinuous equilibrium solutions and cav itation in nonlinear elasticity, Philos

J.M. Ball, Discontinuous equilibrium solutions and cav itation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557-611

work page 1982

[8] [8]

Baroni, M

P. Baroni, M. Colombo, G. Mingione, Nonautonomous funct ionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), no. 3, 347-379

work page 2016

[9] [9]

Baroni, M

P. Baroni, M. Colombo, G. Mingione, Regularity for gener al functionals with double phase, Calc. Var. (2018), 57:62

work page 2018

[10] [10]

Berestycki, P.L

H. Berestycki, P.L. Lions, Nonlinear scalar ﬁeld equat ions, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-345

work page 1983

[11] [11]

Cencelj, V.D

M. Cencelj, V.D. R˘ adulescu, D.D. Repovˇ s, Double phas e problems with variable growth, Nonlinear Anal. 177 (2018), part A, 270-287

work page 2018

[12] [12]

Chorﬁ, V.D

N. Chorﬁ, V.D. R˘ adulescu, Standing wave solutions of a quasilinear degenerate Schr¨ odinger equation with unbounded potential, Electron. J. Qual. Theory Diﬀer. Equ. 2016, Paper No. 37, 12 pp

work page 2016

[13] [13]

De Filippis, Higher integrability for constrained m inimizers of integral functionals with (p, q)-growth in low dimension, Nonlinear Anal

C. De Filippis, Higher integrability for constrained m inimizers of integral functionals with (p, q)-growth in low dimension, Nonlinear Anal. 170 (2018), 1-20

work page 2018

[14] [14]

DiBenedetto, C1+α-local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl

E. DiBenedetto, C1+α-local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl. 7 (1983), 827-850

work page 1983

[15] [15]

Filippucci, P

R. Filippucci, P. Pucci, V.D. R˘ adulescu, Existence an d non-existence results for quasilinear elliptic exterior problems with nonlinear boundary condit ions, Comm. Partial Diﬀerential Equations 33 (2008), 706-717

work page 2008

[16] [16]

Fuˇ cik, J

S. Fuˇ cik, J. Neˇ cas, J. Souˇ cek, V. Souˇ cek,Spectral Analysis of Nonlinear Operators , Lecture Notes in Mathematics, vol. 346, Springer-Verlag, Berlin-N ew York, 1973

work page 1973

[17] [17]

Marcellini, On the deﬁnition and the lower semiconti nuity of certain quasiconvex integrals, Ann

P. Marcellini, On the deﬁnition and the lower semiconti nuity of certain quasiconvex integrals, Ann. Inst. H. Poincar´ e, Anal. Non Lin´ eaire3 (1986), 391-409

work page 1986

[18] [18]

Marcellini, Regularity and existence of solutions o f elliptic equations with p, q–growth conditions, J

P. Marcellini, Regularity and existence of solutions o f elliptic equations with p, q–growth conditions, J. Diﬀerential Equations 90 (1991), 1-30

work page 1991

[19] [19]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s, Dou ble-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), no. 4, Art. 108, 21 pp

work page 2018

[20] [20]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s,Nonlinear Analysis – Theory and Methods , Springer Monographs in Mathematics, Springer, Cham, 2019

work page 2019

[21] [21]

Pucci, J

P. Pucci, J. Serrin, The strong maximum principle revis ited, J. Diﬀerential Equations 196 (2004), no. 1, 1-66

work page 2004

[22] [22]

Pucci, J

P. Pucci, J. Serrin, The Maximum Principle , Progress in Nonlinear Diﬀerential Equations and their Applications, Vol. 73, Birkh¨ auser, Basel, 2007

work page 2007

[23] [23]

Pucci, J

P. Pucci, J. Serrin, Maximum principles for elliptic pa rtial diﬀerential equations, Handbook of diﬀerential equations: stationary partial diﬀerential equations, Vol. IV, 355-483, Handb. Diﬀer. Equ., Elsevier/North-Holland, Amsterdam, 2007

work page 2007

[24] [24]

Pucci, R

P. Pucci, R. Servadei, On weak solutions for p-Laplacian equations with weights, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 18 (2007), no. 3, 257-267

work page 2007

[25] [25]

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

work page 2016

[26] [26]

R˘ adulescu, D

V. R˘ adulescu, D. Repovˇ s, Partial Diﬀerential Equations with Variable Exponents: Va ria- tional Methods and Qualitative Analysis , CRC Press, Taylor & Francis Group, Boca Raton FL, 2015

work page 2015

[27] [27]

Yu, Nonlinear p-Laplacian problems on unbounded domains, Proc

L.S. Yu, Nonlinear p-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc. 115 (1992), 1037-1045

work page 1992

[28] [28]

Zhikov, Averaging of functionals of the calculus o f variations and elasticity theory, Izv

V.V. Zhikov, Averaging of functionals of the calculus o f variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675-710; English translation in Math. USSR- Izv. 29 (1987), no. 1, 33-66. EIGENV ALUE PROBLEMS WITH UNBALANCED GROWTH 13 (I. Farag´ o)MTA-EL TE NumNet Research Group, Budapest, Hungary & Department of Applied Analysi...

work page 1986