We study a double phase Dirichlet problem with a reaction that has a parametric singular term. Using the Nehari manifold method, we show that for all small values of the parameter, the problem has at least two positive, energy minimizing solutions.
Zhenhai Liu
,
Nikolaos S. Papageorgiou
. SINGULAR DOUBLE PHASE EQUATIONS*[J]. Acta mathematica scientia, Series B, 2023
, 43(3)
: 1031
-1044
.
DOI: 10.1007/s10473-023-0304-3
[1] Baroni P, Colombo M, Mingione G. Regularity for general functionals with double phase. Calc Var2019, 57: Art 62
[2] Brown K J, Wu T F.A fibering map approach to a semilinear ellptic boundary value problem. Electr J Diff Equ, 2007, 69(2007): 1-9
[3] Brown K J, Zhang Y.The Nehari manifold for a semilinear elliptic equation with a sign changing weight function. J Differental Equ, 2003, 193: 481-499
[4] Clarke F. Functional Analysis, Calculus of Variations and Optimal Control. London: Springer, 2013
[5] Colasuonno F, Squassina M.Eigenvalues for double phase variational integrals. Annali Mat Pura Appl 2016, 195: 1917-1959
[6] Gasiński L, Papageorgiou N S.Constant sign and nodal solutions for superlinear double phas problems. Adv Calc Var, 2021, 14(4): 613-626
[7] Gasiński L, Winkert P.Sign changing solution for a double phase problems with nonlinear boundary condition via the Nehari manifold. J Differential Equ, 2020, 274: 1037-1066
[8] Ge B, Cao X F, Yuan W S.Existence of two solutions for double-phase problems with a small perturbation. Appl Anal, 2021. doi:10.1060/00036811.2021.1909725
[9] Harjulehto P, Hüasto P.Orlicz Spaces and Generalized Orlicz Spaces. Switzerland: Springer, 2019
[10] Liu W, Dai G.Existence and multiplicity results for double phase problems. J Differential Equ, 2018, 265: 4311-4334
[11] Liu Z, Papageorgiou N S.Double phase Dirichlet problems with unilateral constraints. J Differential Equ, 2022, 316: 249-269
[12] Liu Z, Papageorgiou N S.Anisotropic (p, q)-equations with competition phenomena. Acta Mathematica Scientia, 2022, 42B(1): 299-322
[13] Marcellini P.Regalarity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch Rat Mech Anal, 1989, 105: 267-284
[14] Marcellini P.Reguarity and existence of solutions of elliptic equations with p, q-growth coditions. J Differential Equ 1991, 90(1): 1-30
[15] Papageorgiou N S, Rădulescu V D, Repovš D.Nonlinear Analysis-Theory and Methods. Switzerland: Springer, 2019
[16] Papageorgiou N S, Rădulescu V D, Repovš D. Nonlinear nonhomogeneous singular problems. Calc Var, 2020, 59: Art 9
[17] Papageorgiou N S, Rădulescu V D, Repovš D.Existence and multiplicity of solutions for double-phase Robin problems. Bull London Math Soc, 2020, 52: 546-560
[18] Papageorgiou N S, Rădulescu V D, Zhang Y.Anisotropic singular double phase Dirichlet problems. Disc Cont Dyn Syst S, 2021, 14(12): 4465-4502
[19] Papageorgiou N S, Vetro C, Vetro F.Multiple solutions for parametric double phase Dirichlet problems. Comm Contemp Math, 2021, 23: 2050006
[20] Papageorgiou N S, Winkert P.Applied Nonlinear Functional Analysis. Berlin: DeGruyter, 2018
[21] Perera K, Squassina M.Existence results for double-phase problems via Morse theory. Comm Contemp Math, 2018, 20: 17500023
[22] Ragusa A M, Tachikawa A.Regularity for minimizers for functionals of double phase with variable exponents. Adv Nonlin Anal, 2020, 9: 710-728
[23] Zhikov V V.Averaging of functionals of the calculus of variations and elasticity. Math USSR-Ieves, 1987, 29: 33-66
[24] Zhikov V V.On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J Math Sci, 2011, 173: 463-570
