Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

A class of double phase problem without Ambrosetti-Rabinowitz-type growth condition: infinitely many solutions
  • Home
  • /
  • A class of double phase problem without Ambrosetti-Rabinowitz-type growth condition: infinitely many solutions
  1. Home /
  2. Archives /
  3. Vol 63, No 2 (June 2024) /
  4. Articles

A class of double phase problem without Ambrosetti-Rabinowitz-type growth condition: infinitely many solutions

Authors

  • Bin Ge https://orcid.org/0000-0001-7246-729X
  • Yuhang Han https://orcid.org/0009-0006-7880-1555
  • Qinghai Cao https://orcid.org/0009-0004-2335-6568
  • Haixin Ren https://orcid.org/0000-0003-3453-5727

DOI:

https://doi.org/10.12775/TMNA.2023.040

Keywords

Double phase problem, variational method, Fountain theorem, concave and convex nonlinearities

Abstract

This paper concerns with a class of double phase problem without Ambrosetti-Rabinowitz-type growth condition. Under reasonable hypotheses, we establish the existence of infinitely many solutions by using the variant fountain theorems due to Zou \cite{71}.

References

A. Bahrouni, V.D. Rǎdulescu, D.D. Repovš, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity 32 (2019), 2481–2495.

P. Baroni, M. Colombo and G. Mingione, Nonautonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), 347–379.

P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations 57 (2018), 1–48.

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), 297–324.

A. Benkirane and M. Sidi El Vally, Variational inequalities in Musielak–Orlicz–Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), 787–811.

S.S. Byun and J. Oh, Global gradient estimates for the borderline case of double phase problems with BMO coefficients in nonsmooth domains, J. Differential Equations 263 (2017), 1643–1693.

S.S. Byun and J. Oh, Regularity results for generalized double phase functionals, Anal. PDE 13 (2020), 1269–1300.

L. Cherfils and Y. Il’yasov, On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9–22.

F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. 195 (2016), 1917–1959.

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Rational Mech. Anal. 215 (2015), 443–496.

M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), 219–273.

A. Crespo-Blanco, L. Gasinski, P. Harjulehto and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differential Equations 323 (2022), 182–228.

G. Cupini, P. Marcellini and E. Mascolo, Local boundedness of minimizers with limit growth conditions, J. Optim. Theory Appl. 166 (2015), 1–22.

C. De Filippis and G. Mingione, Manifold constrained non-uniformly elliptic problems, J. Geom. Anal. 30 (2020), 1661–1723.

C. De Filippis and G. Mingione, A borderline case of Calderón–Zygmund estimates for nonuniformly elliptic problems, St. Petersburg Math. J. 31 (2020), 455–477.

X. Fan, An imbedding theorem for Musielak–Sobolev spaces, Nonlinear Anal. 75 (2012), 1959–1971.

L. Gasinski and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations 268 (2020), 4183–4193.

L. Gasinski and P. Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal. 19 (2020), 111739.

L. Gasinski and P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differential Equations 274 (2021), 1037–1066.

B. Ge, X.F. Cao and W.S. Yuan, Existence of two solutions for double-phase problems with a small perturbation, Appl. Anal. 100 (2021), 2147–2162.

B. Ge, D.J. Lv and J.F. Lu, Multiple solutions for a class of double phase problem without the Ambrosetti–Rabinowitz conditions, Nonlinear Anal. 188 (2019), 294–315.

B. Ge, W.S. Yuan and X.F. Cao, Existence and nonexistence of solutions for the double phase problem, Results Math. 76 (2021), 132.

P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Springer, Cham, 2019.

W.L. Liu and G.W. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations 265 (2018), 4311–4334.

W.L. Liu and G.W. Dai, Three ground state solutions for double phase problem, J. Math. Phys. 59 (2018), 121503.

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. vol. 1034, Springer, Berlin, 1983.

J. Ok, Partial regularity for general systems of double phase type with continuous coefficients, Nonlinear Anal. 177 (2018), 673–698.

J. Ok, Regularity for double phase problems under additional integrability assumptions, Nonlinear Anal. 194 (2020), 111408.

N.S. Papageorgiou, V.D. Rădulescu and D. Repovš, Double-phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc. 147 (2019), 2899–2910.

N.S. Papageorgiou, V.D. Rădulescu and D. Repovš, Positive solutions for nonlinear Neumann problems with singular terms and convection, J. Math. Pures Appl. 136 (2020), 1–21.

N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš, Ground state and nodal solutions for a class of double phase problems, Z. Angew. Math. Phys. 71 (2020), 1–15.

K. Perera and M. Squassina, Existence results for double-phase problems via Morse theory, Commun. Contemp. Math. 20 (2018), 1750023.

B.S. Wang, G.L. Hou and B. Ge, Existence of solutions for double phase problems by topological degree, J. Fixed Point Theory Appl. 23(2021), 1–11.

S.D. Zeng, Y.R. Bai, L. Gasinski and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differential Equations 59 (2020), 1–18.

S.D. Zeng, Y.R. Bai, L. Gasinski and P. Winkert, Convergence analysis for double phase obstacle problems with multivalued convection term, Adv. Nonlinear Anal. 10 (2021), 659–672.

V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 675–710.

V.V. Zhikov, On Lavrentiev’s phenomenon, Russ. J. Math. Phys. 3 (1995), 249–269.

V.V. Zhikov, On some variational problems, Russ. J. Math. Phys. 5 (1997), 105–116.

V.V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. 173 (2011), 463–570.

V.V. Zhikov, S.M. Kozlov and O.A. Oleı̌nik, Homogenization of Differential Operators and Integral Functionals, Springer–Verlag, Berlin, 1994.

W.M. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001), 343–358.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2024-06-16

How to Cite

1.
GE, Bin, HAN, Yuhang, CAO, Qinghai and REN, Haixin. A class of double phase problem without Ambrosetti-Rabinowitz-type growth condition: infinitely many solutions. Topological Methods in Nonlinear Analysis. Online. 16 June 2024. Vol. 63, no. 2, pp. 733 - 748. [Accessed 29 June 2025]. DOI 10.12775/TMNA.2023.040.
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 63, No 2 (June 2024)

Section

Articles

License

Copyright (c) 2024 Bin Ge, Yuhang Han, Qinghai Cao, Haixin Ren

Creative Commons License

This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop