A class of double phase variable exponent energy functionals with different power growth and logarithmic perturbation

Yasi Lu; Calogero Vetro; Shengda Zeng

doi:10.3934/dcdss.2024143

Article Contents

2026, Volume 22: 85-121. Doi: 10.3934/dcdss.2024143

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A class of double phase variable exponent energy functionals with different power growth and logarithmic perturbation

1.
College of Mathematics and Information Science, Guangxi University, Nanning, 530004, China
2.
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
3.
National Center for Applied Mathematics in Chongqing and School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

^*Corresponding author: Shengda Zeng
^*Corresponding author: Shengda Zeng

Received: June 2024

Revised: July 2024

Early access: August 13, 2024

Published online: August 13, 2024

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Abstract

In this paper, we study a new kind of double phase variational exponents energy functional with log $ L $-perturbed term, determined by the operator

$ \begin{align} u\mapsto\Delta_{\mathcal {H}_{L}} u = \text{div} \left(\frac{\mathcal {H}'_{L}(x, |\nabla u|)}{|\nabla u|}\nabla u \right), \end{align} $

where

$ \begin{align} \mathcal {H}_{L}(x, t) = [t^{p(x)}+\mu(x)t^{q(x)}]\log(e+\alpha t)\mbox{ for all }~~ x\in \Omega , \mbox{all} ~~ t\ge 0 , \mbox{some }~~ \alpha \geq 0 . \end{align} $

The features of the related Musielak-Orlicz Sobolev spaces are delivered. We further prove the density of smooth functions in such spaces in the case when the domain is bounded, or unbounded but with certain requirements. Finally, under very general assumptions on data, we show the existence and uniqueness results for weak solutions to a special class of perturbed Dirichlet double phase problems.

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Mathematics Subject Classification: Primary: 35B65, 35J60, 35J92.

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[1]	K. S. Albalawi, N. H. Alharthi and F. Vetro, Gradient and parameter dependent Dirichlet $(p(x), q(x))$-Laplace type problem, Mathematics, 10 (2022), 1336. doi: 10.3390/math10081336.
[2]	R. Arora, Á. Crespo-Blanco and P. Winkert, On logarithmic double phase problems, (2023), arXiv preprint, arXiv: 2309.09174.
[3]	A. Bahrouni, V. D. Rădulescu and D. D. Repovš, A weighted anisotropic variant of the Caffarelli-KohnNirenberg inequality and applications, Nonlinearity, 30 (2018), 1516. doi: 10.1088/1361-6544/aaa5dd.
[4]	A. Bahrouni, V. D. Rădulescu and D. D. Repovš, Double phase transonic flow problems with variable growth: Nonlinear patterns and stationary waves, Nonlinearity, 32 (2019), 2481-2495. doi: 10.1088/1361-6544/ab0b03.
[5]	G. Barletta, A. Cianchi and G. Marino, Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces, Calc. Var. Partial Differ. Equ., 62 (2023), Paper No. 65, 42 pp. doi: 10.1007/s00526-022-02393-3.
[6]	P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal., 121 (2015), 206-222. doi: 10.1016/j.na.2014.11.001.
[7]	P. Baroni, M. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersb. Math. J., 27 (2016), 347-379. doi: 10.1090/spmj/1392.
[8]	P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differ. Equ., 57 (2018), Paper No. 62, 48 pp. doi: 10.1007/s00526-018-1332-z.
[9]	L. Beck and G. Mingione, Lipschitz bounds and nonuniform ellipticity, Commun. Pure Appl. Math., 73 (2020), 944-1034. doi: 10.1002/cpa.21880.
[10]	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. doi: 10.1007/s002050000101.
[11]	V. I. Bogachev, Measure Theory, Vol. 1. Berlin: Springer, 2007. doi: 10.1007/978-3-540-34514-5.
[12]	S.-S. Byun and J. Oh, Regularity results for generalized double phase functionals, Anal. PDE, 13 (2020), 1269-1300. doi: 10.2140/apde.2020.13.1269.
[13]	S.-S. Byun, J. Ok and K. Song, Hölder regularity for weak solutions to nonlocal double phase problems, J. Math. Pures Appl., 168 (2022), 110-142. doi: 10.1016/j.matpur.2022.11.001.
[14]	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. doi: 10.3934/cpaa.2005.4.9.
[15]	F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl., 195 (2016), 1917-1959. doi: 10.1007/s10231-015-0542-7.
[16]	M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273. doi: 10.1007/s00205-015-0859-9.
[17]	M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496. doi: 10.1007/s00205-014-0785-2.
[18]	Á. Crespo-Blanco, L. Gasiński, P. Harjulehto and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differ. Equ., 323 (2022), 182-228. doi: 10.1016/j.jde.2022.03.029.
[19]	C. De Filippis and G. Mingione, Lipschitz bounds and nonautonomous integrals, Arch. Ration. Mech. Anal., 242 (2021), 973-1057. doi: 10.1007/s00205-021-01698-5.
[20]	C. De Filippis and G. Mingione, Regularity for double phase problems at nearly linear growth, Arch. Ration. Mech. Anal., 247 (2023), Paper No. 85, 50 pp. doi: 10.1007/s00205-023-01907-3.
[21]	C. De Filippis and G. Palatucci, Hölder regularity for nonlocal double phase equations, J. Differ. Equ., 267 (2019), 547-586. doi: 10.1016/j.jde.2019.01.017.
[22]	L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, 2011. doi: 10.1007/978-3-642-18363-8.
[23]	X. Fan, An imbedding theorem for Musielak-Sobolev spaces, Nonlinear Anal., 75 (2012), 1959-1971. doi: 10.1016/j.na.2011.09.045.
[24]	X. Fan, Boundary trace embedding theorems for variable exponent Sobolev spaces, J. Math. Anal. Appl., 339 (2008), 1395-1412. doi: 10.1016/j.jmaa.2007.08.003.
[25]	X. Fan, Differential equations of divergence form in Musielak-Sobolev spaces and a sub-supersolution method, J. Math. Anal. Appl., 386 (2012), 593-604. doi: 10.1016/j.jmaa.2011.08.022.
[26]	X. Fan, Sobolev embeddings for unbounded domain with variable exponent having values across N, Math. Inequal. Appl., 13 (2010), 123-134. doi: 10.7153/mia-13-09.
[27]	X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k, p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760. doi: 10.1006/jmaa.2001.7618.
[28]	X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446. doi: 10.1006/jmaa.2000.7617.
[29]	M. Fuchs and G. Mingione, Full $C^{1, \alpha}$-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth, Manuscr. Math., 102 (2000), 227-250. doi: 10.1007/s002291020227.
[30]	M. Fuchs and G. Seregin, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Lecture Notes in Mathematics, 1749. Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103751.
[31]	L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differ. Equ., 268 (2020), 4183-4193. doi: 10.1016/j.jde.2019.10.022.
[32]	B. Ge and W. S. Yuan, Quasilinear double phase problems with parameter dependent performance on the whole space, Bulletin Sci. Math., 191 (2024), 103371, 19 pp. doi: 10.1016/j.bulsci.2023.103371.
[33]	P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Springer International Publishing, 2019. doi: 10.1007/978-3-030-15100-3.
[34]	P. Hästö and J. Ok, Maximal regularity for local minimizers of non-autonomous functionals, J. Eur. Math. Soc., 24 (2021), 1285-1334. doi: 10.4171/jems/1118.
[35]	J. Heinonen, T. Kipeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Courier Dover Publications, 2018.
[36]	K. Ho, Y.-H. Kim, P. Winkert and C. Zhang, The boundedness and Hölder continuity of solutions to elliptic equations involving variable exponents and critical growth, J. Differ. Equ., 313 (2022), 503-532. doi: 10.1016/j.jde.2022.01.004.
[37]	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 Differ. Equ., 62 (2023), Paper No. 227, 38 pp. doi: 10.1007/s00526-023-02566-8.
[38]	T. Huo, L. Lin and V. D. Rǎdulescu, Existence and concentration properties for the 1-biharmonic equation with lack of compactness, Bulletin Sci. Math., 186 (2023), 103275, 37 pp. doi: 10.1016/j.bulsci.2023.103275.
[39]	O. Kováčik and J. Rákosník, On spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, Czech. Math. J., 41 (1991), 592-618. doi: 10.21136/CMJ.1991.102493.
[40]	A. L$\hat{e}$, Eigenvalue problems for the p-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099. doi: 10.1016/j.na.2005.05.056.
[41]	W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differ. Equ., 265 (2018), 4311-4334. doi: 10.1016/j.jde.2018.06.006.
[42]	W. L. Liu, G. W. Dai and P. Winkert, Multiple sign-changing solutions for superlinear ($p, q$)-equations in symmetrical expanding domains, Bulletin Sci. Math., 191 (2024), 103393, 21 pp. doi: 10.1016/j.bulsci.2024.103393.
[43]	P. Marcellini, Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, J. Differ. Equ., 90 (1991), 1-30. doi: 10.1016/0022-0396(91)90158-6.
[44]	P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284. doi: 10.1007/BF00251503.
[45]	P. Marcellini and G. Papi, Nonlinear elliptic systems with general growth, J. Differ. Equ., 221 (2006), 412-443. doi: 10.1016/j.jde.2004.11.011.
[46]	J. Musielak, Orlicz Spaces and Modular Spaces, Vol. 1034. Springer, 2006.
[47]	A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(\mathbb{R})$, Math. Inequal. Appl., 7 (2004), 255-266. doi: 10.7153/mia-07-28.
[48]	N. S. Papageorgiou and V. D. Rǎdulescu, An infinity of nodal solutions for superlinear Robin problems with an indefinite and unbounded potential, Bulletin Sci. Math., 141 (2017), 251-266. doi: 10.1016/j.bulsci.2017.03.001.
[49]	V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents, Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015.
[50]	M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal., 9 (2019), 710-728. doi: 10.1515/anona-2020-0022.
[51]	A. Verde, Calderón-Zygmund estimates for systems of $\varphi$-growth, J. Convex Anal., 18 (2011), 67-84.
[52]	C. Vetro and S. Zeng, Regularity and Dirichlet problem for double phase energy functionals of different power growth, J. Geometric Anal., 34 (2024), Paper No. 105, 27 pp. doi: 10.1007/s12220-024-01545-5.
[53]	E. Zeidler, Nonlinear Functional Analysis and its Applications: II/B: Nonlinear monotone operators, Springer Science & Business Media, 2013.
[54]	S. Zeng, Y. Bai, L. Gasiński and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differ. Equ., 59 (2020), Paper No. 176, 18 pp. doi: 10.1007/s00526-020-01841-2.
[55]	S. Zeng, V. D. Rădulescu and P. Winkert, Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions, SIAM J. Math. Anal., 54 (2022), 1898-1926. doi: 10.1137/21M1441195.
[56]	S. Zeng, V. D. Rădulescu and P. Winkert, Nonlocal double phase implicit obstacle problems with multivalued boundary conditions, SIAM J. Math. Anal., 56 (2024), 877-912. doi: 10.1137/22M1501040.
[57]	W. Zhang, J. Zhang and V. D. Rădulescu, Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction, J. Differ. Equ., 347 (2023), 56-103. doi: 10.1016/j.jde.2022.11.033.
[58]	V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.
[59]	V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci., 173 (2011), 463-570. doi: 10.1007/s10958-011-0260-7.
[60]	J. Zuo, W. Q. Zhang and V. D. Rǎdulescu, Multiplicity and concentration properties for ($p, q$)-Kirchhoff non-autonomous problems with Choquard nonlinearity, Bulletin Sci. Math., 191 (2024), 103398, 35 pp. doi: 10.1016/j.bulsci.2024.103398.