Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions

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

doi:10.3934/cam.2024024

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 509-527. doi: 10.3934/cam.2024024

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Research article

Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions

Department of Mathematics, LaMAO, Faculty of Sciences, University Mohammed 1st, Oujda, Morocco

Received: 06 March 2024 Revised: 22 May 2024 Accepted: 22 May 2024 Published: 15 July 2024
35A01, 35J20, 35J60, 35J66

In this article, we study a double phase variable exponents problem with mixed boundary value conditions of the form
$ \left\lbrace \begin{aligned} D(u) +\vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2}u & = f(x,u) \ \ \ \ \text{ in } \Omega,\\ u& = 0 \quad \quad \quad \ \text{ on } \Lambda _1, \\ \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu & = g(x,u) \quad \ \text{ on } \Lambda _2 . \end{aligned} \right. $
First of all, using the mountain pass theorem, we establish that this problem admits at least one nontrivial weak solution without assuming the Ambrosetti–Rabinowitz condition. In addition, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain theorem with the Cerami condition.
- double phase problems,
- variational methods,
- critical point theory,
- Cerami condition
Citation: Mahmoud El Ahmadi, Mohammed Barghouthe, Anass Lamaizi, Mohammed Berrajaa. Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions[J]. Communications in Analysis and Mechanics, 2024, 16(3): 509-527. doi: 10.3934/cam.2024024

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Abstract

In this article, we study a double phase variable exponents problem with mixed boundary value conditions of the form

$ \left\lbrace \begin{aligned} D(u) +\vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2}u & = f(x,u) \ \ \ \ \text{ in } \Omega,\\ u& = 0 \quad \quad \quad \ \text{ on } \Lambda _1, \\ \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu & = g(x,u) \quad \ \text{ on } \Lambda _2 . \end{aligned} \right. $

First of all, using the mountain pass theorem, we establish that this problem admits at least one nontrivial weak solution without assuming the Ambrosetti–Rabinowitz condition. In addition, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain theorem with the Cerami condition.

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
[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.
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[19]	J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034, Springer, Berlin, 1983. https://doi.org/10.1007/BFb0072210
[20]	D. E. Edmunds, J. Rakosnik, Density of smooth functions in $W^{k, p(x)}(\Omega)$, 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
[21]	G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A., 112 (1978), 332–336.
[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
[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
[24]	M. Willem, Minimax Theorems, Birkhauser, Basel, 1996. https://doi.org/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
[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

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