Double-phase problems and a discontinuity property of the spectrum

Papageorgiou, Nikolaos; Rădulescu, Vicenţiu; Repovš, Dušan

doi:10.1090/proc/14466

Double-phase problems and a discontinuity property of the spectrum
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by Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu and Dušan D. Repovš PDF

Proc. Amer. Math. Soc. 147 (2019), 2899-2910 Request permission

Abstract:

We consider a nonlinear eigenvalue problem driven by the sum of $p$ and $q$-Laplacians. We show that the problem has a continuous spectrum. Our result reveals a discontinuity property for the spectrum of a parametric ($p,q$)-differential operator as the parameter $\beta$ goes to $1^-$.

References

John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI 10.1007/BF00279992
P. Baroni, M. Colombo, and G. Mingione, Nonautonomous functionals, borderline cases and related function classes, Algebra i Analiz 27 (2015), no. 3, 6–50; English transl., St. Petersburg Math. J. 27 (2016), no. 3, 347–379. MR 3570955, DOI 10.1090/spmj/1392
Paolo Baroni, Maria Colombo, and Giuseppe Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Paper No. 62, 48. MR 3775180, DOI 10.1007/s00526-018-1332-z
V. Benci, P. D’Avenia, D. Fortunato, and L. Pisani, Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Arch. Ration. Mech. Anal. 154 (2000), no. 4, 297–324. MR 1785469, DOI 10.1007/s002050000101
Nejmeddine Chorfi and Vicenţiu D. Rădulescu, Continuous spectrum for some classes of $(p,2)$-equations with linear or sublinear growth, Miskolc Math. Notes 17 (2016), no. 2, 817–826. MR 3626919, DOI 10.18514/MMN.2017.2170
Maria Colombo and Giuseppe Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273. MR 3360738, DOI 10.1007/s00205-015-0859-9
Cristiana De Filippis, Higher integrability for constrained minimizers of integral functionals with $(p,q)$-growth in low dimension, Nonlinear Anal. 170 (2018), 1–20. MR 3765553, DOI 10.1016/j.na.2017.12.007
Luca Esposito, Francesco Leonetti, and Giuseppe Mingione, Sharp regularity for functionals with $(p,q)$ growth, J. Differential Equations 204 (2004), no. 1, 5–55. MR 2076158, DOI 10.1016/j.jde.2003.11.007
Leszek Gasiński and Nikolaos S. Papageorgiou, Nonlinear analysis, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2168068
Leszek Gasiński and Nikolaos S. Papageorgiou, Multiplicity of positive solutions for eigenvalue problems of $(p,2)$-equations, Bound. Value Probl. , posted on (2012), 2012:152, 17. MR 3016686, DOI 10.1186/1687-2770-2012-152
Leszek Gasiński and Nikolaos S. Papageorgiou, Exercises in analysis. Part 2. Nonlinear analysis, Problem Books in Mathematics, Springer, Cham, 2016. MR 3524637, DOI 10.1007/978-3-319-27817-9
Leszek Gasiński and Nikolaos S. Papageorgiou, Asymmetric $(p,2)$-equations with double resonance, Calc. Var. Partial Differential Equations 56 (2017), no. 3, Paper No. 88, 23. MR 3658341, DOI 10.1007/s00526-017-1180-2
Shouchuan Hu and Nikolas S. Papageorgiou, Handbook of multivalued analysis. Vol. I, Mathematics and its Applications, vol. 419, Kluwer Academic Publishers, Dordrecht, 1997. Theory. MR 1485775, DOI 10.1007/978-1-4615-6359-4
Gary M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural′tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2-3, 311–361. MR 1104103, DOI 10.1080/03605309108820761
Salvatore A. Marano, Sunra J. N. Mosconi, and Nikolaos S. Papageorgiou, Multiple solutions to $(p,q)$-Laplacian problems with resonant concave nonlinearity, Adv. Nonlinear Stud. 16 (2016), no. 1, 51–65. MR 3456746, DOI 10.1515/ans-2015-5011
Salvatore A. Marano, Sunra J. N. Mosconi, and Nikolaos S. Papageorgiou, On a $(p,q)$-Laplacian problem with parametric concave term and asymmetric perturbation, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 1, 109–125. MR 3787723, DOI 10.4171/RLM/796
Paolo Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 391–409 (English, with French summary). MR 868523
Paolo Marcellini, Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions, J. Differential Equations 90 (1991), no. 1, 1–30. MR 1094446, DOI 10.1016/0022-0396(91)90158-6
Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), no. 3, 393–430. MR 3197304, DOI 10.1007/s00245-013-9227-z
Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu, Noncoercive resonant $(p,2)$-equations, Appl. Math. Optim. 76 (2017), no. 3, 621–639. MR 3715857, DOI 10.1007/s00245-016-9363-3
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, and Dušan D. Repovš, Existence and multiplicity of solutions for resonant $(p,2)$-equations, Adv. Nonlinear Stud. 18 (2018), no. 1, 105–129. MR 3748157, DOI 10.1515/ans-2017-0009
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, and Dušan D. Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), no. 4, Paper No. 108, 21. MR 3836199, DOI 10.1007/s00033-018-1001-2
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, and Dušan D. Repovš, $(p,2)$-equations asymmetric at both zero and infinity, Adv. Nonlinear Anal. 7 (2018), no. 3, 327–351. MR 3836120, DOI 10.1515/anona-2017-0195
N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš, Perturbations of nonlinear eigenvalue problems, Comm. Pure Appl. Anal. 18 (2019), 1403–1431.
Vicenţiu D. Rădulescu, Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal. 121 (2015), 336–369. MR 3348928, DOI 10.1016/j.na.2014.11.007
Vicenţiu D. Rădulescu and Dušan D. Repovš, Partial differential equations with variable exponents, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015. Variational methods and qualitative analysis. MR 3379920, DOI 10.1201/b18601
Honghui Yin and Zuodong Yang, A class of $p$-$q$-Laplacian type equation with concave-convex nonlinearities in bounded domain, J. Math. Anal. Appl. 382 (2011), no. 2, 843–855. MR 2810836, DOI 10.1016/j.jmaa.2011.04.090
V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877 (Russian). MR 864171
Vasiliĭ V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3 (1995), no. 2, 249–269. MR 1350506