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Isotropic and anisotropic double-phase problems: old and new

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Języki publikacji
EN
Abstrakty
EN
We are concerned with the study of two classes of nonlinear problems driven by differential operators with unbalanced growth, which generalize the (p, q)- and (p(x),q(x))-Laplace operators. The associated energy is a double-phase functional, either isotropic or anisotropic. The content of this paper is in relationship with pioneering contributions due to P. Marcelimi and G. Mingione.
Rocznik
Strony
259--279
Opis fizyczny
Bibliogr. 34 poz.
Twórcy
  • AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Kraków, Poland
  • Institute of Mathematics Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia
  • Institute of Mathematics "Simion Stoilow" Romanian Academy of Sciences P.O. Box 1-764, 014700 Bucharest, Romania
Bibliografia
  • [1] E. Acerbi, G. Mingione, Gradient estimates for the p{x)-Laplacean system,, J. Reine Angew. Math. 584 (2005), 117-148.
  • [2] A. Azzollini, Minimum action solutions for a quasilinear equation, J. London Math. Soc. 92 (2015), 583-595.
  • [3] A. Azzollini, P. d'Avenia, A. Pomponio, Quasilinear elliptic equations in M,N via varia-tional methods and Orlicz-Sobolev ernbeddings, Calc. Var. Partial Differential Equations 49 (2014), 197-213.
  • [4] M. Badiale, L. Pisani, S. Rolando, Sum of weighted Lebesgue spaces and nonlinear elliptic equations, Nonlinear Differ. Equ. Appl. (NoDEA) 18 (2011), 369-405.
  • [5] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A 306 (1982) 1496, 557-611.
  • [6] S. Baraket, S. Chebbi, N. Chorfi, V.D. Radulescu, Non-autonomous eigenvalue problems with variable (pi,p2~)-growth, Advanced Nonlinear Studies 17 (2017), 781-792.
  • [7] P. Baroni, M. Colombo, G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206-222.
  • [8] P. Baroni, M. Colombo, G. Mingione, Nonautonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016) 3, 347-379.
  • [9] P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations (2018), 57:62.
  • [10] H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-345.
  • [11] H. Brezis, F. Browder, Partial differential equations in the 20th century, Adv. Math. 135 (1998), 76-144.
  • [12] H. Brezis, L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), 939-963.
  • [13] J. Byeon, Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrodinger equations, Arch. Ration. Mech. Anal. 165 (2002), 295-316.
  • [14] M. Cencelj, V.D. Radulescu, D.D. Repovs, Double phase problems with variable growth, Nonlinear Anal. 177 (2018), 270-287.
  • [15] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (2006), 1383-1406.
  • [16] N. Chorfi, V.D. Radulescu, Standing wave solutions of a quasilinear degenerate Schrodinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 37 (2016), 1-12.
  • [17] M. Colombo, G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015) 2, 443-496.
  • [18] M. Colombo, G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015) 1, 219-273.
  • [19] C. De Filippis, Higher integrability for constrained minimizers of integral functionals with (p,q)-growth in low dimension, Nonlinear Anal. 170 (2018), 1-20.
  • [20] L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with (p,q)-growth, J. Differential Equations 204 (2004), 5-55.
  • [21] S. Fućik, J. Nećas, J. Soućek, V. Soućek, Spectral Analysis of Nonlinear Operators, Lecture Notes in Mathematics, vol. 346, Springer-Verlag, Berlin-New York, 1973.
  • [22] I.H. Kim, Y.H. Kim, Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math. 147 (2015), 169-191.
  • [23] P. Marcelimi, On the definition and the lower semicontinuity of certain quasi-convex integrals, Ann. Inst. H. Poincare, Anal. Non Lineaire 3 (1986), 391-409.
  • [24] P. Marcelimi, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991), 1-30.
  • [25] P. Marcelimi, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996) 1, 1-25.
  • [26] G. Mingione, Talk at the Third Conference on Recent Trends in Nonlinear Phenomena, University of Perugia, 28-30 September 2016.
  • [27] N.S. Papageorgiou, V.D. Radulescu, D.D. Repovs, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), Art. 108, 21 pp.
  • [28] P. Pucci, V.D. Radulescu, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. Ser. IX 3 (2010), 543-584.
  • [29] P. Pucci, S. Saldi, Critical stationary Kirchhoff equations in M,N involving nonlocal operators, Rev. Mat. Iberoam. 32 (2016), 1-22.
  • [30] P. Pucci, M. Xiang, B. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal. 5 (2016), 27-55.
  • [31] V.D. Radulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Analysis: Theory, Methods and Applications 121 (2015), 336-369.
  • [32] V.D. Radulescu, D.D. Repovs, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
  • [33] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986) 4, 675-710; English translation in Math. USSR-Izv. 29 (1987) 1, 33-66.
  • [34] V.V. Zhikov, On Lavrentiev's phenomenon, Russian J. Math. Phys. 3 (1995) 2, 249-269.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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