Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting. (English) Zbl 1534.35182
Summary: This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti-Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35J66 | Nonlinear boundary value problems for nonlinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
References:
| [1] | Baroni, P.; Colombo, M.; Mingione, G., Harnack inequalities for double phase functionals, Nonlinear Anal., 121, 206-222 (2015) · Zbl 1321.49059 · doi:10.1016/j.na.2014.11.001 |
| [2] | Baroni, P.; Colombo, M.; Mingione, G., Regularity for general functionals with double phase, Calc. Var. Partial Differ. Equ., 57, 62, 48 (2018) · Zbl 1394.49034 |
| [3] | Bartolo, T.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7, 241-273 (1983) · Zbl 0522.58012 · doi:10.1016/0362-546X(83)90115-3 |
| [4] | Brézis, H., Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext (2011), New York: Springer, New York · Zbl 1220.46002 · doi:10.1007/978-0-387-70914-7 |
| [5] | Colasuonno, F.; Squassina, M., Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4), 195, 1917-1959 (2016) · Zbl 1364.35226 · doi:10.1007/s10231-015-0542-7 |
| [6] | Colombo, M.; Mingione, G., Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218, 219-273 (2015) · Zbl 1325.49042 · doi:10.1007/s00205-015-0859-9 |
| [7] | Colombo, FM; Mingione, G., Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215, 443-496 (2015) · Zbl 1322.49065 · doi:10.1007/s00205-014-0785-2 |
| [8] | Crespo-Blanco, Á.; Gasiński, L.; Harjulehto, P.; Winkert, P., A new class of double phase variable exponent problems: existence and uniqueness, J. Differ. Equ., 323, 182-228 (2022) · Zbl 1489.35041 · doi:10.1016/j.jde.2022.03.029 |
| [9] | Diening, L., Harjulehto, P., Hästö, P., Ru̇žička, M.: Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011) · Zbl 1222.46002 |
| [10] | El Manouni, S.; Marino, G.; Winkert, P., Existence results for double phase problems depending on Robin and Steklov eigenvalues for the \(p\)-Laplacian, Adv. Nonlinear Anal., 11, 1, 304-320 (2022) · Zbl 1479.35487 · doi:10.1515/anona-2020-0193 |
| [11] | Farkas, C.; Fiscella, A.; Winkert, P., Singular Finsler double phase problems with nonlinear boundary condition, Adv. Nonlinear Stud., 21, 4, 809-825 (2021) · Zbl 1485.35155 · doi:10.1515/ans-2021-2143 |
| [12] | Farkas, C.; Winkert, P., An existence result for singular Finsler double phase problems, J. Differ. Equ., 286, 455-473 (2021) · Zbl 1465.35148 · doi:10.1016/j.jde.2021.03.036 |
| [13] | Fiscella, A., A double phase problem involving Hardy potentials, Appl. Math. Optim., 85, 45, 16 (2022) · Zbl 1498.35270 |
| [14] | Fiscella, A.; Pinamonti, A., Existence and multiplicity results for Kirchhoff type problems on a double phase setting, Mediterr. J. Math., 20, 33, 19 (2023) · Zbl 1511.35171 |
| [15] | Gasiński, L.; Papageorgiou, NS, Constant sign and nodal solutions for superlinear double phase problems, Adv. Calc. Var., 14, 613-626 (2021) · Zbl 1478.35118 · doi:10.1515/acv-2019-0040 |
| [16] | Gasiński, L.; Winkert, P., Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal., 195, 111739 (2020) · Zbl 1437.35233 · doi:10.1016/j.na.2019.111739 |
| [17] | Gasiński, L.; Winkert, P., Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differ. Equ., 274, 1037-1066 (2021) · Zbl 1458.35149 · doi:10.1016/j.jde.2020.11.014 |
| [18] | Ge, B.; Lv, DJ; Lu, JF, Multiple solutions for a class of double phase problem without the Ambrosetti-Rabinowitz conditions, Nonlinear Anal., 188, 294-315 (2019) · Zbl 1429.35085 · doi:10.1016/j.na.2019.06.007 |
| [19] | Lê, A., Eigenvalue problems for the \(p\)-Laplacian, Nonlinear Anal., 64, 5, 1057-1099 (2006) · Zbl 1208.35015 · doi:10.1016/j.na.2005.05.056 |
| [20] | Liu, S., On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73, 788-795 (2010) · Zbl 1192.35074 · doi:10.1016/j.na.2010.04.016 |
| [21] | Liu, W.; Dai, G., Existence and multiplicity results for double phase problem, J. Differ. Equ., 265, 4311-4334 (2018) · Zbl 1401.35103 · doi:10.1016/j.jde.2018.06.006 |
| [22] | Marcellini, P., Regularity of minimisers of integrals of the calculus of variations with non standard growth conditions, Arch. Ration. Mech. Anal., 105, 267-284 (1989) · Zbl 0667.49032 · doi:10.1007/BF00251503 |
| [23] | Marcellini, P., Regularity and existence of solutions of elliptic equations with \((p, q)\)-growth conditions, J. Differ. Equ., 90, 1-30 (1991) · Zbl 0724.35043 · doi:10.1016/0022-0396(91)90158-6 |
| [24] | Musielak, J.: Orlicz spaces and modular spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983) · Zbl 0557.46020 |
| [25] | Papageorgiou, NS; Rădulescu, VD; Repovš, DD, Ground state and nodal solutions for a class of double phase problems, Z. Angew. Math. Phys., 71, 1-15 (2020) · Zbl 1437.35358 · doi:10.1007/s00033-019-1239-3 |
| [26] | Perera, K.; Squassina, M., Existence results for double-phase problems via Morse theory, Commun. Contemp. Math., 20, 14 (2018) · Zbl 1379.35152 · doi:10.1142/S0219199717500237 |
| [27] | Willem, M., Minimax Theorems (1996), Basel: Birkhäuser, Basel · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1 |
| [28] | Zhikov, VV, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50, 675-710 (1986) |
| [29] | Zhikov, VV, On Lavrentiev’s phenomenon, Russ. J. Math. Phys., 3, 249-269 (1995) · Zbl 0910.49020 |
| [30] | Zhikov, VV, On some variational problems, Russ. J. Math. Phys., 5, 105-116 (1997) · Zbl 0917.49006 |
| [31] | Zhikov, VV; Kozlov, SM; Oleinik, OA, Homogenization of Differential Operators and Integral Functionals (1994), Berlin: Springer, Berlin · Zbl 0838.35001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
