Opuscula Math. 39, no. 2 (2019), 259-279
https://doi.org/10.7494/OpMath.2019.39.2.259

Opuscula Mathematica

# Isotropic and anisotropic double-phase problems: old and new

Vicenţiu D. Rădulescu

Abstract. 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. Marcellini and G. Mingione.

Keywords: differential operator with unbalanced growth, double-phase energy, variable exponent.

Mathematics Subject Classification: 35J60, 35J65, 58E05.

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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 $$\mathbb{R}^N$$ via variational methods and Orlicz-Sobolev embeddings, 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. Rădulescu, Non-autonomous eigenvalue problems with variable $$(p_1,p_2)$$-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 Schrödinger equations, Arch. Ration. Mech. Anal. 165 (2002), 295-316.
14. M. Cencelj, V.D. Rădulescu, D.D. Repovš, 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. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger 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. Fučik, J. Nečas, J. Souček, V. Souček, 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. Marcellini, On the definition and the lower semicontinuity of certain quasi-convex integrals, Ann. Inst. H. Poincaré, Anal. Non Linéaire 3 (1986), 391-409.
24. P. Marcellini, Regularity and existence of solutions of elliptic equations with $$p,q$$-growth conditions, J. Differential Equations 90 (1991), 1-30.
25. P. Marcellini, 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. Rădulescu, D.D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), Art. 108, 21 pp.
28. P. Pucci, V.D. Rădulescu, 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 $$\mathbb{R}^N$$ 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. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Analysis: Theory, Methods and Applications 121 (2015), 336-369.
32. V.D. Rădulescu, D.D. Repovš, 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.
• Vicenţiu D. Rădulescu
• https://orcid.org/0000-0003-4615-5537
• 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
• Communicated by Dušan Repovš.
• Accepted: 2018-11-10.
• Published online: 2018-12-07.