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El Ahmadi, Mahmoud; Barghouthe, Mohammed; Lamaizi, Anass; Berrajaa, Mohammed

Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions. (English) Zbl 07920446

Commun. Anal. Mech. 16, No. 3, 509-527 (2024).

MSC:

35Jxx Elliptic equations and elliptic systems
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J66 Nonlinear boundary value problems for nonlinear elliptic equations

Keywords:

double phase problems; variational methods; critical point theory; Cerami condition
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Full Text: DOI

References:

[1] V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Berlin: Springer-Verlag, 1994. https://doi.org/10.1007/978-3-642-84659-5 · Zbl 0838.35001 · doi:10.1007/978-3-642-84659-5
[2] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 50 (1986), 675-710.
[3] V. V. Zhikov, On Lavrentiev’s phenomenon, Russian journal of mathematical physics, 3 (1995), 2. · Zbl 0910.49020
[4] P. Baroni, M. Colombo, G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222. https://doi.org/10.1016/j.na.2014.11.001 doi: 10.1016/j.na.2014.11.001 · Zbl 1321.49059 · doi:10.1016/j.na.2014.11.001
[5] P. Baroni, M. Colombo, G. Mingione, Regularity for general functionals with double phase, Calc. Var., 57 (2018), 1-48. https://doi.org/10.1007/s00526-018-1332-z doi: 10.1007/s00526-018-1332-z · Zbl 1394.49034 · doi:10.1007/s00526-018-1332-z
[6] P. Baroni, T. Kuusi, G. Mingione, Borderline gradient continuity of minima, J. Fixed Point Theory Appl., 15 (2014), 537-575. https://doi.org/10.1007/s11784-014-0188-x doi: 10.1007/s11784-014-0188-x · Zbl 1308.49027 · doi:10.1007/s11784-014-0188-x
[7] G. Cupini, P. Marcellini, E. Mascolo, Local boundedness of minimizers with limit growth conditions, J. Optim. Theory Appl., 166 (2015), 1-22. https://doi.org/10.1007/s10957-015-0722-z doi: 10.1007/s10957-015-0722-z · Zbl 1325.49043 · doi:10.1007/s10957-015-0722-z
[8] W. Liu, G. Dai, Existence and multiplicity results for double phase problem, J. Differ. Equations, 265 (2018), 4311-4334. https://doi.org/10.1016/j.jde.2018.06.006 doi: 10.1016/j.jde.2018.06.006 · Zbl 1401.35103 · doi:10.1016/j.jde.2018.06.006
[9] M. El Ahmadi, M. Berrajaa, A. Ayoujil, Existence of two solutions for Kirchhoff double phase problems with a small perturbation without (AR)-condition, Discrete Contin. Dyn. Syst. S. https://doi.org/10.3934/dcdss.2023085 · doi:10.3934/dcdss.2023085
[10] L. Gasiński, P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differ. Equations, 274 (2021), 1037-1066. https://doi.org/10.1016/j.jde.2020.11.014 doi: 10.1016/j.jde.2020.11.014 · Zbl 1458.35149 · doi:10.1016/j.jde.2020.11.014
[11] N. Cui, H. R. Sun, Existence and multiplicity results for double phase problem with nonlinear boundary condition, Nonlinear Anal.: Real World Appl., 60 (2021), 103307. https://doi.org/10.1016/j.nonrwa.2021.103307 doi: 10.1016/j.nonrwa.2021.103307 · Zbl 1466.35200 · doi:10.1016/j.nonrwa.2021.103307
[12] Y. Yang, W. Liu, P. Winkert, X. Yan, Existence of solutions for resonant double phase problems with mixed boundary value conditions, Partial Differ. Equ. Appl., 4 (2023), 18. https://doi.org/10.1007/s42985-023-00237-z doi: 10.1007/s42985-023-00237-z · Zbl 1518.35280 · doi:10.1007/s42985-023-00237-z
[13] Z. Liu, S. Zeng, L. Gasiński, Y. H. Kim, Nonlocal double phase complementarity systems with convection term and mixed boundary conditions, J. Geom. Anal., 32 (2022), 241. https://doi.org/10.1007/s12220-022-00977-1 doi: 10.1007/s12220-022-00977-1 · Zbl 1497.35142 · doi:10.1007/s12220-022-00977-1
[14] A. Benkirane, M. Sidi El Vally, Variational inequalities in Musielak-Orlicz-Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin, 21 (2014), 787-811. https://doi.org/10.36045/bbms/1420071854 doi: 10.36045/bbms/1420071854 · Zbl 1326.35142 · doi:10.36045/bbms/1420071854
[15] F. Colasuonno, M. Squassina, Eigenvalues for double phase variational integrals, Annali di Matematica, 195 (2016), 1917-1959. https://doi.org/10.1007/s10231-015-0542-7 doi: 10.1007/s10231-015-0542-7 · Zbl 1364.35226 · doi:10.1007/s10231-015-0542-7
[16] Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differ. Equations, 323 (2022), 182-228. https://doi.org/10.1016/j.jde.2022.03.029 doi: 10.1016/j.jde.2022.03.029 · Zbl 1489.35041 · doi:10.1016/j.jde.2022.03.029
[17] X. Fan, An imbedding theorem for Musielak-Sobolev spaces, Nonlinear Anal., 75 (2012), 1959-1971. https://doi.org/10.1016/j.na.2011.09.045 doi: 10.1016/j.na.2011.09.045 · Zbl 1272.46024 · doi:10.1016/j.na.2011.09.045
[18] P. Harjulehto, P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Lecture Notes in Mathematics, 2236, Springer Cham, 2019. https://doi.org/10.1007/978-3-030-15100-3 · Zbl 1436.46002 · doi:10.1007/978-3-030-15100-3
[19] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034, Springer, Berlin, 1983. https://doi.org/10.1007/BFb0072210 · Zbl 0557.46020 · doi:10.1007/BFb0072210
[20] D. E. Edmunds, J. Rakosnik, Density of smooth functions in Wk,p(x)(Ω), Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 437 (1992), 229-236. https://doi.org/10.1098/rspa.1992.0059 doi: 10.1098/rspa.1992.0059 · Zbl 0779.46027 · doi:10.1098/rspa.1992.0059
[21] G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A., 112 (1978), 332-336. · Zbl 0436.58006
[22] D. Motreanu, V. V. Motreanu, N.S Papageorgiou, Topological and variational methods with applications to nonlinear boundary value problems, New York: Springer, 2014. https://doi.org/10.1007/978-1-4614-9323-5 · Zbl 1292.47001 · doi:10.1007/978-1-4614-9323-5
[23] A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7 · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[24] M. Willem, Minimax Theorems, Birkhauser, Basel, 1996. https://doi.org/10.1007/978-1-4612-4146-1 · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1
[25] M. El Ahmadi, A. Ayoujil, M. Berrajaa, Existence and multiplicity of solutions for a class of double phase variable exponent problems with nonlinear boundary condition, Adv. Math. Models Appl., 8 (2023), 401-414.
[26] Q. Zhang, C. Zhao, Existence of strong solutions of a p(x)-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comput. Math. Appl., 69 (2015), 1-12. https://doi.org/10.1016/j.camwa.2014.10.022 doi: 10.1016/j.camwa.2014.10.022 · Zbl 1360.35022 · doi:10.1016/j.camwa.2014.10.022
[27] A. Ayoujil, On the superlinear Steklov problem involving the p(x)-Laplacian, Electron. J. Qual. Theory Differ. Equations, 2014 (2014), 1-13. https://doi.org/10.14232/ejqtde.2014.1.38 doi: 10.14232/ejqtde.2014.1.38 · Zbl 1324.35046 · doi:10.14232/ejqtde.2014.1.38
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