Infinitely many positive solutions for a double phase problems involving the double phase operator
DOI:
https://doi.org/10.12775/TMNA.2024.013Keywords
Double phase operator, oscillatory nonlinearities, infinitely many positive solutions, variational principleAbstract
In this paper we study a double phase problem which involves the double phase operator, and the nonlinear term has an oscillatory behavior. By using variational methods and the theory of the Musielak-Orlicz-Sobolev space, we establish the existence of infinitely many solutions whose $W_0^{1,H}(\Omega)$-norms tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity, respectively).References
A. Benkirane and M. Sidi El Vally, Variational inequalities in Musielak–Orlicz–Sobolev spaces, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), 787–811.
M. Cencelj, V.D. Radulescu and D.D. Repovs, Double phase problems with variable growth, Nonlinear Anal. 177 (2018), 270–287.
F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. 195 (2016), 1917–1959.
A. Crespo-Blanco, L. Gasinski, P. Harjulehto and P. Winkert, Anewclass of double phase variable exponent problems: existence and uniqueness, J. Differential Equations 323 (2022), 182–228.
G. Cuesta and G. Figueiredo, Existence and concentration of positive solutions for a critical p&q equation, Adv. Nonlinear Anal. 11 (2022), 243–267.
X. Fan and C.X. Guan, Uniform convexity of Musielak–Orlicz–Sobolev spaces and applications, Nonlinear Anal. 73 (2010), 163–175.
B. Ge and Z.Y. Chen, Existence of infinitely many solutions for double phase problem with sign-changing potential, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM. 113 (2019), 3185–3196.
B. Ge, D.L. 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, L.Y. Wang and J.F. Lu, On a class of double-phase problem without Ambrosetti–Robinowitz-type conditions, Appl. Anal. 100 (2021), 2147–2162.
K. Ho and P. Winkert, New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems, Calc. Var. Partial Differential Equations 62 (2023), 227.
J. Liu and P. Pucci, Existence of solutions for a double-phase variable exponent equation without the Ambrosetti–Rabinowitz condition, Adv. Nonlinear Anal. 12 (2023), 20220292.
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.
M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217–229.
J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer, Berlin, 1983.
N.S. Papageorgiou, Double phase problems: a survey of some recent results, Opuscula Math. 42 (2022), 257–278.
N.S. Papageorgiou, A. Pudelko and V.D. Radulescu, Non-autonomous (p, q)equations with unbalanced growth, Math. Ann., 385 (2023), 1707–1745.
N.S. Papageorgiou, D.D. Qin and V.D. Radulescu, Nonlinear eigenvalue problems for the (p, q)-Laplacian, Bull. Sci. Math. 172 (2021), 103039.
N.S. Papageorgiou, V.D. Radulescu and D.D. Repovs, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69 (2018), 108.
N.S. Papageorgiou, V.D. Radulescu and D. Repovs, Global multiplicity for parametric anisotropic Neumann (p, q)-equations, Topol. Methods Nonlinear Anal. 61 (2023), 393–422.
N.S. Papageorgiou, D.D. Repovs and V. D. Radulescu, Double-phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc. 147 (2019), 2899–2910.
V.D. Radulescu, Isotropic and anisotropic double-phase problems: old and new, Opuscula Math. 39 (2019), 259–279.
Q.H. Zhang and V.D. Radulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. 118 (2018), 159–203.
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, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer–Verlag, Berlin, 1994.
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