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Overview of differential equations with non-standard growth. (English) Zbl 1188.35072
Summary: Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey, we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.

MSC:
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] L. Diening, P. Hästö, A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: P. Drábek, J. Rákosník (Eds.), FSDONA04 Proceedings, Milovy, Czech Republic, 2004, pp. 38-58.
[2] Samko, S., On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral transforms spec. funct., 16, 5-6, 461-482, (2005) · Zbl 1069.47056
[3] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Book manuscript, 2010.
[4] Mingione, G., Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. math., 51, (2006) · Zbl 1164.49324
[5] X.-L., Fan, \(p(x)\)-Laplacian equations, (), 117-123 · Zbl 1205.35115
[6] Orlicz, W., Über konjugierte exponentenfolgen, Studia math., 3, 200-212, (1931) · JFM 57.0251.02
[7] Nakano, H., Modulared semi-ordered linear spaces, (1950), Maruzen Co., Ltd. Tokyo · Zbl 0041.23401
[8] Nakano, H., Topology and topological linear spaces, (1951), Maruzen Co., Ltd. Tokyo
[9] Musielak, J., Orlicz spaces and modular spaces, (1983), Springer-Verlag Berlin · Zbl 0557.46020
[10] Zhikov, V.V., Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR izv., 29, 1, 33-66, (1987), (Translation of Izv. Akad. Nauk SSSR Ser. Mat. 50 (4) (1986) 675-710, 877) · Zbl 0599.49031
[11] Kováčik, O., Some properties of the spaces \(L^{p(t)}(\Omega)\), (), 98-100, 267 (in Russian)
[12] Kováčik, O., Parabolic equations in generalized Sobolev spaces \(W^{k, p(x)}\), Fasc. math., 25, 87-94, (1995) · Zbl 0840.35040
[13] Fan, X.-L., The regularity of Lagrangians \(f(x, \xi) = | \xi |^{\alpha(x)}\) with Hölder exponents \(\alpha(x)\), Acta math. sinica (N.S.), 12, 3, 254-261, (1996) · Zbl 0874.49031
[14] Fan, X.-L., Regularity of nonstandard Lagrangians \(f(x, \xi)\), Nonlinear anal., 27, 6, 669-678, (1996) · Zbl 0874.49032
[15] Fan, X.-L.; Zhao, D., Regularity of minimizers of variational integrals with continuous \(p(x)\)-growth conditions, Chinese J. contemp. math., 17, 4, 327-336, (1996)
[16] Alkhutov, Yu.A., The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with nonstandard growth condition, Differ. uravn., 33, 12, 1651-1660, (1997), 1726. (Translation in Differ. Equ. 33 (12) (1997) 1653-1663) · Zbl 0949.35048
[17] Marcellini, P., Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions, J. differential equations, 90, 1, 1-30, (1991) · Zbl 0724.35043
[18] Růžička, M., Flow of shear dependent electrorheological fluids: unsteady space periodic case, (), 485-504 · Zbl 0954.35138
[19] Růžička, M., Electrorheological fluids: modeling and mathematical theory, (2000), Springer-Verlag Berlin · Zbl 0968.76531
[20] Acerbi, E.; Mingione, G., Regularity results for stationary electro-rheological fluids, Arch. ration. mech. anal., 164, 213-259, (2002) · Zbl 1038.76058
[21] Acerbi, E.; Mingione, G.; Seregin, G., Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. inst. H. Poincaré anal. non linéaire, 21, 1, 25-60, (2004) · Zbl 1052.76004
[22] Antontsev, S.N.; Rodrigues, J.F., On stationary thermo-rheological viscous flows, Ann. univ. ferrara sez. VII sci. mat., 52, 1, 19-36, (2006) · Zbl 1117.76004
[23] Chen, Y.; Levine, S.; Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. appl. math., 66, 4, 1383-1406, (2006) · Zbl 1102.49010
[24] Aboulaich, R.; Meskine, D.; Souissi, A., New diffusion models in image processing, Comput. math. appl., 56, 4, 874-882, (2008) · Zbl 1155.35389
[25] Bollt, E.; Chartrand, R.; Esedog¯lu, S.; Schultz, P.; Vixie, K., Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion, Adv. comput. math., 31, 1-3, 61-85, (2009) · Zbl 1169.94302
[26] A. El Hamidi, C. Ghannam, G. Bailly-Maitre, M. Menard, Nonstandard diffusion in image restoration and decomposition, Preprint, 2009.
[27] Levine, S., An adaptive variational model for image decomposition, (), 382-397
[28] Eleuteri, M.; Habermann, J., Regularity results for a class of obstacle problems under nonstandard growth conditions, J. math. anal. appl., 344, 2, 1120-1142, (2008) · Zbl 1147.49028
[29] M. Eleuteri, J. Habermann, A Hölder continuity result for a class of obstacle problems under non standard growth conditions, Math. Nachr. (in press). · Zbl 1225.35073
[30] M. Eleuteri, J. Habermann, Calderón-Zygmund type estimates for a class of obstacle problems with \(p(x)\) growth, Preprint, 2010. · Zbl 1211.49046
[31] Harjulehto, P.; Hästö, P.; Koskenoja, M.; Lukkari, T.; Marola, N., An obstacle problems and superharmonic functions with nonstandard growth, Nonlinear anal., 67, 3424-3440, (2007) · Zbl 1130.31004
[32] Rodrigues, J.F.; Sanchón, M.; Urbano, J.M., The obstacle problem for nonlinear elliptic equations with variable growth and \(L^1\)-data, Monatsh. math., 154, 4, 303-322, (2008) · Zbl 1155.35051
[33] Antontsev, S.; Shmarev, S., A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear anal., 60, 3, 515-545, (2005) · Zbl 1066.35045
[34] Henriques, E.; Urbano, J.M., Intrinsic scaling for PDE’s with an exponential nonlinearity, Indiana univ. math. J., 55, 5, 1701-1721, (2006) · Zbl 1105.35023
[35] Dai, G.; Hao, R., Existence of solutions for a \(p(x)\)-Kirchhoff-type equation, J. math. anal. appl., 359, 1, 275-284, (2009) · Zbl 1172.35401
[36] Dai, G.; Liu, D., Infinitely many positive solutions for a \(p(x)\)-Kirchhoff-type equation, J. math. anal. appl., 359, 2, 704-710, (2009) · Zbl 1173.35463
[37] Autuori, G.; Pucci, P.; Salvatori, M.C., Asymptotic stability for anisotropic Kirchhoff systems, J. math. anal. appl., 352, 1, 149-165, (2009) · Zbl 1175.35013
[38] G. Autuori, P. Pucci, M.C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal. (in press). · Zbl 1201.35138
[39] Diening, L.; Růžička, M., Calderón – zygmund operators on generalized Lebesgue spaces \(L^{p(\cdot)}\) and problems related to fluid dynamics, J. reine angew. math., 563, 197-220, (2003) · Zbl 1072.76071
[40] Pinasco, Juan P., Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear anal., 71, 3-4, 1094-1099, (2009) · Zbl 1170.35341
[41] Calotă, L., On some quasilinear elliptic equations with critical Sobolev exponents and non-standard growth conditions, Bull. belg. math. soc. Simon stevin, 15, 2, 249-256, (2008) · Zbl 1157.35039
[42] Garcia Melian, J.; Rossi, J.D.; Sabina, J., Existence, asymptotic behavior and uniqueness for large solutions to \(\operatorname{\Delta} u = \operatorname{e}^{q(x) u}\), Adv. nonlinear stud., 9, 395-424, (2009) · Zbl 1185.35092
[43] Garcia Melian, J.; Rossi, J.D.; Sabina, J., Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. inst. H. Poincaré anal. non linéaire, 26, 889-902, (2009) · Zbl 1177.35072
[44] Zhang, Q.H., Existence and asymptotic behavior of blow-up solutions to a class of \(p(x)\)-Laplacian problems, J. math. anal. appl., 329, 1, 472-482, (2007) · Zbl 1241.35075
[45] Zhang, Q.-H., Boundary blow-up solutions to \(p(x)\)-Laplacian equations with exponential nonlinearities, J. inequal. appl., (2008), Article ID 279306, 8 pp
[46] Zhang, Q.H.; Liu, X.P.; Qiu, Z.M., On the boundary blow-up solutions of \(p(x)\)-Laplacian equations with singular coefficient, Nonlinear anal., 70, 11, 4053-4070, (2009) · Zbl 1165.35387
[47] Habermann, J., Partial regularity for minima of higher order functionals with \(p(x)\)-growth, Manuscripta math., 126, 1, 1-40, (2008) · Zbl 1142.49017
[48] Habermann, J., Calderón – zygmund estimates for higher order systems with \(p(x)\) growth, Math. Z., 258, 2, 427-462, (2008) · Zbl 1147.42004
[49] Cianci, P.; Nicolosi, F., Boundedness of solutions for an elliptic equation with non-standard growth, Nonlinear anal., 71, 5-6, 1825-1832, (2009) · Zbl 1170.35396
[50] T. Adamowicz, P. Hästö, Mappings of finite distortion and PDE with nonstandard growth, Int. Math. Res. Not. IMRN (2010), in press (doi:10.1093/imrn/rnp192). · Zbl 1206.35134
[51] Bonder, J.; Martínez, S.; Wolanski, N., A free boundary problem for the \(p(x)\)-Laplacian, Nonlinear anal., 72, 2, 1078-1103, (2010) · Zbl 1183.35096
[52] Harjulehto, P., Variable exponent Sobolev spaces with zero boundary values, Math. bohem., 132, 2, 125-136, (2007) · Zbl 1174.46322
[53] Heinonen, J.; Kilpeläinen, T.; Martio, O., ()
[54] Kinderlehrer, D.; Stampacchia, G., ()
[55] Malý, J.; Ziemer, W.P., ()
[56] Harjulehto, P.; Hästö, P.; Koskenoja, M., The Dirichlet energy integral on intervals in variable exponent Sobolev spaces, Z. anal. anwend., 22, 4, 911-923, (2003) · Zbl 1046.46027
[57] Fan, X.-L.; Fan, X., A knobloch-type result for \(p(t)\)-Laplacian systems, J. math. anal. appl., 282, 453-464, (2003) · Zbl 1033.34023
[58] Wang, X.-J.; Yuan, R., Existence of periodic solutions for \(p(t)\)-Laplacian systems, Nonlinear anal., 70, 2, 866-880, (2009) · Zbl 1171.34030
[59] Fan, X.-L.; Wu, H.-Q.; Wang, F.-Z., Hartman-type results for \(p(t)\)-Laplacian systems, Nonlinear anal., 52, 585-594, (2003) · Zbl 1025.34017
[60] Zhang, Q.-H., Oscillatory property of solutions for \(p(t)\)-Laplacian equations, J. inequal. appl., (2007), Article ID 58548, 8 pp
[61] Zhang, Q.H.; Qiu, Z.M.; Liu, X.P., Existence of multiple solutions for weighted \(p(r)\)-Laplacian equation Dirichlet problems, Nonlinear anal., 70, 10, 3721-3729, (2009) · Zbl 1180.34018
[62] Zhang, Q.H.; Liu, X.P.; Qiu, Z.M., Existence of solutions and nonnegative solutions for weighted \(p(r)\)-Laplacian impulsive system multi-point boundary value problems, Nonlinear anal., 71, 9, 3814-3825, (2009) · Zbl 1183.34037
[63] Zhang, Q.H.; Qiu, Z.M.; Liu, X.P., Existence of solutions for weighted \(p(r)\)-Laplacian impulsive system periodic-like boundary value problems, Nonlinear anal., 71, 9, 3596-3611, (2009) · Zbl 1178.34036
[64] Hästö, P., On the variable exponent Dirichlet energy integral, Commun. pure appl. anal., 5, 3, 413-420, (2006)
[65] Harjulehto, P.; Hästö, P.; Koskenoja, M.; Varonen, S., The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential anal., 25, 3, 205-222, (2006) · Zbl 1120.46016
[66] Harjulehto, P.; Hästö, P.; Latvala, V., Minimizers of the variable exponent, non-uniformly convex Dirichlet energy, J. math. pures appl. (9), 89, 2, 174-197, (2008) · Zbl 1142.49007
[67] Harjulehto, P.; Hästö, P.; Latvala, V., Harnack’s inequality for \(p(\cdot)\)-harmonic functions with unbounded exponent \(p\), J. math. anal. appl., 352, 1, 345-359, (2009) · Zbl 1204.46021
[68] Manfredi, J.J.; Rossi, J.D.; Urbano, J.M., \(p(x)\)-harmonic function with unbounded exponent in a subdomain, Ann. inst. H. Poincaré anal. non linéaire, 26, 2581-2595, (2009) · Zbl 1180.35242
[69] P. Lindqvist, T. Lukkari, A curious equation involving the infinity-Laplacian, Adv. Calc. Var. 2009 (in press).
[70] Manfredi, J.J.; Rossi, J.D.; Urbano, J.M., Limits as \(p(x) \rightarrow \infty\) of \(p(x)\)-harmonic functions, Nonlinear anal., 72, 309-315, (2010) · Zbl 1181.35118
[71] Perez-Llanos, M.; Rossi, J., The behaviour of the \(p(x)\)-Laplacian eigenvalue problem as \(p(x) \rightarrow \operatorname{infinity}\), J. math. anal. appl., 363, 2, 502-511, (2010) · Zbl 1182.35176
[72] Coscia, A.; Mucci, D., Integral representation and \(\Gamma\)-convergence of variational integrals with \(p(x)\)-growth, ESAIM control optim. calc. var., 7, 495-519, (2002) · Zbl 1036.49022
[73] Galewska, E.; Galewski, M., On the stability of solutions for the \(p(x)\)-Laplacian equation and some applications to optimisation problems with state constraints, Anziam j., 48, 2, 245-257, (2006) · Zbl 1156.35378
[74] Fan, X.-L.; Zhang, Q.-H.; Zhao, D., Eigenvalues of \(p(x)\)-Laplacian Dirichlet problem, J. math. anal. appl., 302, 306-317, (2005) · Zbl 1072.35138
[75] Allegretto, W., Form estimates for the \(p(x)\)-Laplacian, Proc. amer. math. soc., 135, 7, 2177-2185, (2007) · Zbl 1123.35032
[76] Fan, X.-L., A constrained minimization problem involving the \(p(x)\)-Laplacian in image, Nonlinear anal., 69, 10, 3661-3670, (2008) · Zbl 1159.35326
[77] Fan, X.-L., Eigenvalues of the \(p(x)\)-Laplacian Neumann problem, Nonlinear anal., 67, 2982-2992, (2007) · Zbl 1126.35037
[78] Deng, S.-G., Eigenvalues of the \(p(x)\)-Laplacian Steklov problem, J. math. anal. appl., 339, 925-937, (2008) · Zbl 1160.49307
[79] Deng, S.-G., A local mountain pass theorem and applications to a double perturbed \(p(x)\)-Laplacian equations, Appl. math. comput., 211, 1, 234-241, (2009) · Zbl 1173.35045
[80] Mihăilescu, M.; Rădulescu, V., On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. amer. math. soc., 135, 9, 2929-2937, (2007) · Zbl 1146.35067
[81] Mihăilescu, M.; Pucci, P.; Rădulescu, V., Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. math. anal. appl., 340, 1, 687-698, (2008) · Zbl 1135.35058
[82] Kurata, K.; Shioji, N., Compact embedding from \(W_0^{1, 2}(\Omega)\) to \(L^{q(x)}(\Omega)\) and its application to nonlinear elliptic boundary value problem with variable critical exponent, J. math. anal. appl., 339, 2, 1386-1394, (2008) · Zbl 1151.46020
[83] Fan, X.-L., Remarks on eigenvalue problems involving the \(p(x)\)-Laplacian, J. math. anal. appl., 352, 1, 85-98, (2009) · Zbl 1163.35026
[84] Zhang, Q.-H., Existence and asymptotic behavior of positive solutions to \(p(x)\)-Laplacian equations with singular nonlinearities, J. inequal. appl., (2007), Article ID 19349, 9 pp
[85] Fan, X.-L.; Zhang, Q.-H., Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem, Nonlinear anal., 52, 1843-1852, (2003) · Zbl 1146.35353
[86] Sanchón, M.; Urbano, J.M., Entropy solutions for the \(p(x)\)-Laplace equation, Trans. amer. math. soc., 361, 12, 6387-6405, (2009) · Zbl 1181.35121
[87] Iliaş, P.S., Existence and multiplicity of solutions of a \(p(x)\)-Laplacian equation in a bounded domain, Rev. roumaine math. pures appl., 52, 6, 639-653, (2007) · Zbl 1174.35009
[88] Chabrowski, J.; Fu, Y., Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain, J. math. anal. appl., 306, 604-618, (2005) · Zbl 1160.35399
[89] Antontsev, S.; Shmarev, S., Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear anal., 65, 4, 728-761, (2006) · Zbl 1245.35033
[90] Antontsev, S.; Shmarev, S., Parabolic equations with anisotropic nonstandard growth conditions, (), 33-44 · Zbl 1113.35098
[91] Antontsev, S.N.; Consiglieri, L., Elliptic boundary value problems with nonstandard growth conditions, Nonlinear anal., 71, 3-4, 891-902, (2009) · Zbl 1175.35048
[92] Dai, G., Infinitely many solutions for a hemivariational inequality involving the \(p(x)\)-Laplacian, Nonlinear anal., 71(, 1-2, 186-195, (2009) · Zbl 1173.35511
[93] Dai, G., Infinitely many solutions for a \(p(x)\)-Laplacian equation in \(\mathbb{R}^N\), Nonlinear anal., 71, 3-4, 1133-1139, (2009) · Zbl 1170.35431
[94] Dai, G., Infinitely many non-negative solutions for a Dirichlet problem involving \(p(x)\)-Laplacian, Nonlinear anal., 71, 11, 5840-5849, (2009) · Zbl 1182.35128
[95] Fan, X.-L., \(p(x)\)-Laplacian equations in \(\mathbb{R}^n\) with periodic data and nonperiodic perturbations, J. math. anal. appl., 341, 1, 103-119, (2008) · Zbl 1135.35034
[96] Fan, X.-L.; Deng, S.-G., Remarks on ricceri’s variational principle and applications to the \(p(x)\)-Laplacian equations, Nonlinear anal., 67, 3064-3075, (2007) · Zbl 1134.35035
[97] Fan, X.-L.; Deng, S.-G., Multiplicity of positive solutions for a class of inhomogeneous Neumann problems involving the \(p(x)\)-Laplacian, Nodea nonlinear differential equations appl., 16, 255-271, (2009) · Zbl 1173.35491
[98] Fan, X.-L.; Han, X., Existence and multiplicity of solutions for \(p(x)\)-Laplacian equations in \(R^N\), Nonlinear anal., 59, 1-2, 173-188, (2004) · Zbl 1134.35333
[99] Fan, X.-L.; Zhao, D., Regularity of quasi-minimizers of integral functionals with discontinuous \(p(x)\)-growth conditions, Nonlinear anal., 65, 8, 1521-1531, (2006) · Zbl 1142.35021
[100] Fan, X.-L.; Zhao, Y., Nodal solutions of \(p(x)\)-Laplacian equations, Nonlinear anal., 67, 10, 2859-2868, (2007) · Zbl 1143.35063
[101] Fu, Y., The principle of concentration compactness in \(L^{p(x)}\) spaces and its application, Nonlinear anal., 71, 5-6, 1876-1892, (2009) · Zbl 1170.35402
[102] Fu, Y.Q.; Zhang, X., A multiplicity result for \(p(x)\)-Laplacian problem in \(\mathbb{R}^N\), Nonlinear anal., 70, 6, 2261-2269, (2009) · Zbl 1156.35363
[103] Iliaş, P.S., Dirichlet problem with \(p(x)\)-Laplacian, Math. rep. (bucur.), 10, 60, 43-56, (2008) · Zbl 1199.35086
[104] Liu, W.; Zhao, P., Existence of positive solutions for \(p(x)\)-Laplacian equations in unbounded domains, Nonlinear anal., 69, 10, 3358-3371, (2008) · Zbl 1158.35360
[105] Mihăilescu, M., Elliptic problems in variable exponent spaces, Bull. austral. math. soc., 74, 2, 197-206, (2006) · Zbl 1284.35177
[106] Yao, J., Solutions for Neumann boundary value problems involving \(p(x)\)-Laplace operators, Nonlinear anal., 68, 1271-1283, (2008) · Zbl 1158.35046
[107] Zhang, Q.-H., Existence of solutions for \(p(x)\)-Laplacian equations with singular coefficients in \(\mathbb{R}^N\), J. math. anal. appl., 348, 1, 38-50, (2008) · Zbl 1156.35035
[108] Zang, A., \(p(x)\)-Laplacian equations satisfying Cerami condition, J. math. anal. appl., 337, 547-555, (2008) · Zbl 1216.35065
[109] Ben Ali, K.; Bezzarga, M., On a nonhomogeneous quasilinear problem in Sobolev spaces with variable exponent, Bul. ştiinţ. univ. piteşti ser. mat. inf., 14, Suppl., 19-38, (2008) · Zbl 1224.35151
[110] Boureanu, M.-M., Existence of solutions for an elliptic equation involving the \(p(x)\)-Laplace operator, Electron. J. differential equations, 97, 1-10, (2006)
[111] Mihăilescu, M., Existence and multiplicity of solutions for a Neumann problem involving the \(p(x)\)-Laplace operator, Nonlinear anal., 67, 5, 1419-1425, (2007) · Zbl 1163.35381
[112] Yao, J.; Wang, X., On an open problem involving the \(p(x)\)-laplacian—A further study on the multiplicity of weak solutions to \(p(x)\)-Laplacian equations, Nonlinear anal., 69, 1445-1453, (2008) · Zbl 1144.35390
[113] Papageorgiou, N.S.; Rocha, E.M., A multiplicity theorem for a variable exponent Dirichlet problem, Glasg. math. J., 50, 2, 335-349, (2008) · Zbl 1154.35041
[114] Mihăilescu, M., Existence and multiplicity of solutions for an elliptic equation with \(p(x)\)-growth conditions, Glasg. math. J., 48, 411-418, (2006) · Zbl 1387.35289
[115] Liu, Q., Existence of three solutions for \(p(x)\)-Laplacian equations, Nonlinear anal., 68, 7, 2119-2127, (2008) · Zbl 1135.35329
[116] Ji, C., Perturbation for a \(p(x)\)-Laplacian equation involving oscillating nonlinearities in \(\mathbb{R}^N\), Nonlinear anal., 69, 2393-2402, (2008) · Zbl 1152.35041
[117] Galewski, M., On the existence and stability of solutions for Dirichlet problem with \(p(x)\)-Laplacian, J. math. anal. appl., 326, 352-362, (2007) · Zbl 1159.35365
[118] Galewski, M., On a Dirichlet problem with generalized \(p(x)\)-Laplacian and some applications, Numer. funct. anal. optim., 28, 1087-1111, (2007) · Zbl 1154.35036
[119] Galewski, M., On a Dirichlet problem with \(p(x)\)-Laplacian, J. math. anal. appl., 337, 281-291, (2008) · Zbl 1132.35030
[120] Alves, C.O., Existence of solution for a degenerate \(p(x)\)-Laplacian equation in \(\mathbb{R}^N\), J. math. anal. appl., 345, 2, 731-742, (2008) · Zbl 1146.35044
[121] Corrêa, F.J.S.A.; Costa, A.C.R.; Figueiredo, G.M., On a singular elliptic problem involving the \(p(x)\)-Laplacian and generalized lebesgue – sobolev spaces, Adv. math. sci. appl., 17, 2, 639-650, (2007) · Zbl 1133.35373
[122] Lukkari, T., Elliptic equations with nonstandard growth involving measures, Hiroshima math. J., 38, 155-176, (2008) · Zbl 1148.35034
[123] Mihăilescu, M., On a class of nonlinear problems involving a \(p(x)\)-Laplace type operator, Czechoslovak math. J., 58, 133, 155-172, (2008) · Zbl 1165.35336
[124] Mihăilescu, M.; Rădulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. soc. lond. ser. A math. phys. eng. sci., 462, 2073, 2625-2641, (2006) · Zbl 1149.76692
[125] Mihăilescu, M.; Rădulescu, V., Continuous spectrum for a class of nonhomogeneous differential operators, Manuscripta math., 125, 2, 157-167, (2008) · Zbl 1138.35070
[126] Mihăilescu, M.; Rădulescu, V.; Repovš, D., On a non-homogeneous eigenvalue problem involving a potential: an orlicz – sobolev space setting, J. math. pures appl., 93, 132-148, (2010) · Zbl 1186.35116
[127] Dinu, T.-L., Nonlinear eigenvalue problems in Sobolev spaces with variable exponent, J. funct. spaces appl., 4, 3, 225-242, (2006) · Zbl 1165.35335
[128] Boureanu, M.M.; Mihăilescu, M., Existence and multiplicity of solutions for a Neumann problem involving variable exponent growth conditions, Glasg. math. J., 50, 565-574, (2008) · Zbl 1188.35089
[129] Rovenţa, I., Boundary asymptotic and uniqueness of solution for a problem with \(p(x)\)-Laplacian, J. inequal. appl., (2008), Article ID 609047, 14 pp · Zbl 1158.35038
[130] Fu, Y.Q., The existence of solutions for elliptic systems with nonuniform growth, Studia math., 151, 3, 227-246, (2002) · Zbl 1007.35023
[131] Diening, L.; Růžička, M., Strong solutions for generalized Newtonian fluids, J. math. fluid mech., 7, 413-450, (2005) · Zbl 1080.76005
[132] Zhang, Q.-H., Existence of positive solutions for elliptic systems with nonstandard \(p(x)\)-growth conditions via sub – supersolution method, Nonlinear anal., 67, 1055-1067, (2007) · Zbl 1166.35326
[133] Zhang, Q.-H., Existence of positive solutions for a class of \(p(x)\)-Laplacian systems, J. math. anal. appl., 333, 2, 591-603, (2007) · Zbl 1154.35032
[134] Zhang, Q.-H., Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems, Nonlinear anal., 70, 1, 305-316, (2009) · Zbl 1161.35376
[135] Afrouzi, G.A.; Ghorbani, H., Existence of positive solutions for \(p(x)\)-Laplacian problems, Electron. J. differential equations, 177, 1-9, (2007) · Zbl 1146.35348
[136] El Hamidi, A., Existence results to elliptic systems with nonstandard growth conditions, J. math. anal. appl., 300, 1, 30-42, (2004) · Zbl 1148.35316
[137] Xu, X.; An, Y., Existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in \(\mathbb{R}^N\), Nonlinear anal., 68, 4, 956-968, (2008) · Zbl 1142.35018
[138] Ogras, S.; Mashiyev, R.A.; Avci, M.; Yucedag, Z., Existence of solutions for a class of elliptic systems in \(\mathbb{R}^N\) involving the \((p(x), q(x))\)-Laplacian, J. inequal. appl., (2008), Art.ID 612938, 16 pp · Zbl 1180.35212
[139] Liu, J.; Shi, X., Existence of three solutions for a class of quasilinear elliptic systems involving the \((p(x), q(x))\)-Laplacian, Nonlinear anal., 71, 1-2, 550-557, (2009) · Zbl 1167.35359
[140] Guo, X.L.; Lu, M.X.; Zhang, Q.H., Infinitely many periodic solutions for variable exponent systems, J. inequal. appl., (2009), Art. No. 714179
[141] Harjulehto, P.; Kinnunen, J.; Lukkari, T., Unbounded supersolutions of nonlinear equations with nonstandard growth, Bound. value probl., (2007), Article ID 48348, 20 pp · Zbl 1161.35020
[142] Alkhutov, Yu.A.; Krasheninnikova, O.V., Continuity at boundary points of solutions of quasilinear elliptic equations with a non-standard growth condition, Izv. ross. akad. nauk ser. mat., 68, 6, 3-60, (2004), (Translation in Izv. Math. 68 (6) (2004) 1063-1117) · Zbl 1167.35385
[143] P. Harjulehto, P. Hästö, V. Latvala, Boundedness of solutions of the non-uniformly convex, non-standard growth Laplacian, Preprint, 2010. · Zbl 1254.35070
[144] Fan, X.-L.; Zhao, Y.Z.; Zhang, Q.-H., A strong maximum principle for \(p(x)\)-Laplace equations, Chinese J. contemp. math., 24, 3, 277-282, (2003), (Translation of Chinese Ann. Math. Ser. A, 24 (4) (2003) 495-500)
[145] Fortini, R.; Mugnai, D.; Pucci, P., Maximum principles for anisotropic elliptic inequalities, Nonlinear anal., 70, 8, 2917-2929, (2009) · Zbl 1169.35314
[146] Hästö, P., Counter-examples of regularity in variable exponent Sobolev spaces, () · Zbl 1084.46025
[147] Zhikov, V.V., On lavrentiev’s phenomenon, Russ. J. math. phys., 3, 249-269, (1995) · Zbl 0910.49020
[148] M. Eleuteri, P. Harjulehto, T. Lukkari, Global regularity and stability of solutions to elliptic equations with nonstandard growth, Complex Var. Elliptic Equ. (in press). · Zbl 1232.35030
[149] Alkhutov, Yu.A., On the Hölder continuity of \(p(x)\)-harmonic functions, Mat. sb., 196, 2, 3-28, (2005), (Translation in Sb. Math. 196 (1-2) (2005) 147-171) · Zbl 1121.35047
[150] Harjulehto, P.; Latvala, V., Fine topology of variable exponent energy superminimizers, Ann. acad. sci. fenn. math., 33, 491-510, (2008) · Zbl 1152.31005
[151] T. Lukkari, F.-Y. Maeda, N. Marola, Wolff potential estimates for elliptic equations with nonstandard growth and applications. Forum Math. (in press). · Zbl 1203.35099
[152] Lukkari, T., Singular solutions of elliptic equations with nonstandard growth, Math. nachr., 282, 12, 1770-1787, (2009) · Zbl 1180.35253
[153] V. Latvala, T. Lukkari, O. Toivanen, The fundamental convergence theorem for \(p(\cdot)\)-superharmonic functions, Preprint, 2009. · Zbl 1228.31005
[154] Pastukhova, S.E.; Zhikov, V.V., Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent, Sb. math., 199, 12, 1751-1782, (2008) · Zbl 1172.35024
[155] Fan, X.-L.; Zhao, D., A class of De Giorgi type and Hölder continuity, Nonlinear anal., 36, 295-318, (1999) · Zbl 0927.46022
[156] Fan, X.-L.; Zhao, D., The quasi-minimizer of integral functionals with \(m(x)\) growth conditions, Nonlinear anal., 39, 807-816, (2000) · Zbl 0943.49029
[157] Acerbi, E.; Mingione, G., Regularity results for a class of functionals with non-standard growth, Arch. ration. mech. anal., 156, 121-140, (2001) · Zbl 0984.49020
[158] Coscia, A.; Mingione, G., Hölder continuity of the gradient of \(p(x)\)-harmonic mappings, C. R. acad. sci. Paris, 328, 363-368, (1999) · Zbl 0920.49020
[159] Giaquinta, M.; Giusti, E., One the regularity of the minima of variational integrals, Acta math., 148, 31-46, (1982) · Zbl 0494.49031
[160] Giaquinta, M.; Giusti, E., Quasiminima, Ann. inst. H. Poincaré anal. non linéaire, 1, 2, 79-107, (1984)
[161] Habermann, J.; Zatorska-Goldstein, A., Regularity for minimizers of functionals with nonstandard growth by \(\mathcal{A}\)-harmonic approximation, Nodea nonlinear differential equations appl., 15, 1-2, 169-194, (2008) · Zbl 1183.49038
[162] Eleuteri, M., Hölder continuity results for a class of functionals with non standard growth, Boll. unione mat. ital., 7-B, 129-157, (2004) · Zbl 1178.49045
[163] Acerbi, E.; Mingione, G., Regularity results for a class of quasiconvex functionals with nonstandard growth, Ann. sc. norm. super. Pisa cl. sci. (4), 30, 2, 311-339, (2001) · Zbl 1027.49031
[164] Acerbi, E.; Mingione, G., Gradient estimates for the \(p(x)\)-Laplacian system, J. reine angew. math., 584, 117-148, (2005) · Zbl 1093.76003
[165] Zhikov, V.V., Meyer-type estimates for solving the nonlinear Stokes system, Differ. equ., 33, 1, 108-115, (1997), (Translation of Differ. Uravn. 33 (1) (1997) 107-114, 143) · Zbl 0911.35089
[166] Zhikov, V.V., On some variational problems, Russ. J. math. phys., 5, 1, 105-116, (1997) · Zbl 0917.49006
[167] Chiadò Piat, V.; Coscia, A., Hölder continuity of minimizers of functionals with variable growth exponent, Manuscripta math., 93, 3, 283-299, (1997) · Zbl 0878.49010
[168] Harjulehto, P.; Kuusi, T.; Lukkari, T.; Marola, N.; Parviainen, M., Harnack’s inequality for quasiminimizers with non-standard growth conditions, J. math. anal. appl., 344, 1, 504-520, (2008) · Zbl 1145.49023
[169] Fan, X.-L., Global \(C^{1, \alpha}\) regularity for variable exponent elliptic equations in divergence form, J. differential equations, 235, 2, 397-417, (2007) · Zbl 1143.35040
[170] T. Lukkari, Boundary continuity of solutions to elliptic equations with nonstandard growth, Preprint, 2009. · Zbl 1180.35253
[171] Li, H.; Wu, Z.; Yin, J.; Zhao, J., Nonlinear diffusion equations, (2001), World Scientific New Jersey
[172] Chen, Y.Z.; Xu, M., Hölder continuity of weak solutions for parabolic equations with nonstandard growth conditions, Acta math. sin. (engl. ser.), 22, 3, 793-806, (2006) · Zbl 1107.35041
[173] DiBenedetto, E., ()
[174] DiBenedetto, E.; Gianazza, U.; Vespri, V., Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta math., 200, 2, 181-209, (2008) · Zbl 1221.35213
[175] Urbano, J.M., ()
[176] Antontsev, S.; Zhikov, V., Higher integrability for parabolic equations of \(p(x, t)\)-Laplacian type, Adv. differential equations, 10, 9, 1053-1080, (2005) · Zbl 1122.35043
[177] M. Nuortio, Local boundedness for parabolic variable exponent PDE, Internationally unpublished Licentiate of Philosophy Thesis, University of Oulu, 2008.
[178] Diening, L.; Ettwein, F.; Růžička, M., \(C^{1, \alpha}\)-regularity for electrorheological fluids in two dimensions, Nodea nonlinear differential equations appl., 14, 1-2, 207-217, (2007) · Zbl 1132.76301
[179] Crispo, F.; Grisanti, C.R., On the \(C^{1, \gamma}(\overline{\Omega}) \cap W^{2, 2}(\Omega)\) regularity for a class of electro-rheological fluids, J. math. anal. appl., 356, 1, 119-132, (2009) · Zbl 1178.35300
[180] Bendahmane, M.; Wittbold, P., Renormalized solutions for nonlinear elliptic equations with variable exponents and \(L^1\) data, Nonlinear anal., 70, 2, 567-583, (2009) · Zbl 1152.35384
[181] Bögelein, V.; Zatorska-Goldstein, A., Higher integrability of very weak solutions of systems of \(p(x)\)-Laplacian type, J. math. anal. appl., 336, 1, 480-497, (2007) · Zbl 1387.35205
[182] Iwaniec, T.; Sbordone, C., Weak minima of variational integrals, J. reine angew. math., 454, 143-161, (1994) · Zbl 0802.35016
[183] Lewis, J.L., On very weak solutions of certain elliptic systems, Comm. partial differential equations, 18, 9-10, 1515-1537, (1993) · Zbl 0796.35061
[184] Bögelein, V., Very weak solutions of higher order degenerate parabolic systems, Adv. differential equations, 14, 1-2, 121-200, (2009) · Zbl 1178.35215
[185] Kinnunen, J.; Lewis, J.L., Very weak solutions of parabolic systems of \(p\)-Laplacian type, Ark. mat., 40, 1, 105-132, (2002) · Zbl 1011.35039
[186] Bénilan, P.; Boccardo, L.; Gallouët, T.; Gariepy, R.; Pierre, M.; Vázquez, J.L., An \(L^1\)-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. sc. norm. super. Pisa cl. sci. (4), 22, 2, 241-273, (1995) · Zbl 0866.35037
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