Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 16, 2018

Multiple solutions of double phase variational problems with variable exponent

  • Xiayang Shi , Vicenţiu D. Rădulescu EMAIL logo , Dušan D. Repovš ORCID logo and Qihu Zhang

Abstract

This paper deals with the existence of multiple solutions for the quasilinear equation

-div𝐀(x,u)+|u|α(x)-2u=f(x,u)in N,

which involves a general variable exponent elliptic operator 𝐀 in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like |ξ|q(x)-2ξ for small |ξ| and like |ξ|p(x)-2ξ for large |ξ|, where 1<α()p()<q()<N. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in N via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponents p and q are constant, to the case where p() and q() are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.

MSC 2010: 35J62; 35J70; 35J20

Communicated by Giuseppe Mingione


Funding statement: Vicenţiu D. Rădulescu and Dušan D. Repovš were supported by the Slovenian Research Agency, grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083.

References

[1] E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2001), no. 2, 121–140. 10.1007/s002050100117Search in Google Scholar

[2] C. O. Alves and S. Liu, On superlinear p(x)-Laplacian equations in N, Nonlinear Anal. 73 (2010), no. 8, 2566–2579. 10.1016/j.na.2010.06.033Search in Google Scholar

[3] S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: Existence, uniqueness and localization properties of solutions, Nonlinear Anal. 65 (2006), no. 4, 728–761. 10.1016/j.na.2005.09.035Search in Google Scholar

[4] A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in N via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 (2014), no. 1–2, 197–213. 10.1007/s00526-012-0578-0Search in Google Scholar

[5] P. Baroni, M. Colombo and G. Mingione, Nonautonomous functionals, borderline cases and related function classes, Algebra i Analiz 27 (2015), no. 3, 6–50. Search in Google Scholar

[6] M.-M. Boureanu and V. D. Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. 75 (2012), no. 12, 4471–4482. 10.1016/j.na.2011.09.033Search in Google Scholar

[7] J. Chabrowski and Y. Fu, Existence of solutions for p(x)-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), no. 2, 604–618. 10.1016/j.jmaa.2004.10.028Search in Google Scholar

[8] K. C. Chang, Critical Point Theory and Applications (in Chinese), Shanghai Scientific and Technology Press, Shanghai, 1986. Search in Google Scholar

[9] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383–1406. 10.1137/050624522Search in Google Scholar

[10] N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 37. 10.14232/ejqtde.2016.1.37Search in Google Scholar

[11] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM, Philadelphia, 2013. Search in Google Scholar

[12] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273. 10.1007/s00205-015-0859-9Search in Google Scholar

[13] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443–496. 10.1007/s00205-014-0785-2Search in Google Scholar

[14] A. Coscia and G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 4, 363–368. 10.1016/S0764-4442(99)80226-2Search in Google Scholar

[15] L. Diening, Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp() and Wk,p(), Math. Nachr. 268 (2004), no. 1, 31–43. 10.1002/mana.200310157Search in Google Scholar

[16] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar

[17] D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267–293. 10.4064/sm-143-3-267-293Search in Google Scholar

[18] A. El Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004), no. 1, 30–42. 10.1016/j.jmaa.2004.05.041Search in Google Scholar

[19] X. Fan, Global C1,α regularity for variable exponent elliptic equations in divergence form, J. Differential Equations 235 (2007), no. 2, 397–417. 10.1016/j.jde.2007.01.008Search in Google Scholar

[20] X. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl. 262 (2001), no. 2, 749–760. 10.1006/jmaa.2001.7618Search in Google Scholar

[21] X. Fan, Q. Zhang and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), no. 2, 306–317. 10.1016/j.jmaa.2003.11.020Search in Google Scholar

[22] X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424–446. 10.1006/jmaa.2000.7617Search in Google Scholar

[23] X.-L. Fan and Q.-H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), no. 8, 1843–1852. 10.1016/S0362-546X(02)00150-5Search in Google Scholar

[24] Y. Fu, The principle of concentration compactness in Lp(x) spaces and its application, Nonlinear Anal. 71 (2009), no. 5–6, 1876–1892. 10.1016/j.na.2009.01.023Search in Google Scholar

[25] Y. Fu and Y. Shan, On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal. 5 (2016), no. 2, 121–132. 10.1515/anona-2015-0055Search in Google Scholar

[26] P. Harjulehto, P. Hästö and V. Latvala, Harnack’s inequality for p()-harmonic functions with unbounded exponent p, J. Math. Anal. Appl. 352 (2009), no. 1, 345–359. 10.1016/j.jmaa.2008.05.090Search in Google Scholar

[27] P. Harjulehto, P. Hästö, V. Latvala and O. Toivanen, Critical variable exponent functionals in image restoration, Appl. Math. Lett. 26 (2013), no. 1, 56–60. 10.1016/j.aml.2012.03.032Search in Google Scholar

[28] P. Harjulehto, P. Hästö, U. V. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551–4574. 10.1016/j.na.2010.02.033Search in Google Scholar

[29] P. Harjulehto, V. Latvala and O. Toivanen, A variant of the Geman–McClure model for image restoration, J. Math. Anal. Appl. 399 (2013), no. 2, 676–681. 10.1016/j.jmaa.2012.10.050Search in Google Scholar

[30] H. Hudzik, On density of C(Ω) in Orlicz–Sobolev space WMk(Ω) for every open set Ωn, Funct. Approx. Comment. Math. 5 (1977), 113–128. Search in Google Scholar

[31] H. Hudzik, On imbedding theorems of Orlicz–Sobolev space WMk(Ω) into Cm(Ω) for open, bounded, and starlike Ωn, Comment. Math. Prace Mat. 20 (1977/78), no. 2, 341–363. Search in Google Scholar

[32] R. C. James, Reflexivity and the sup of linear functionals, Israel J. Math. 13 (1972), no. 3–4, 289–300. 10.1007/BF02762803Search in Google Scholar

[33] T. Kopaliani, Interpolation theorems for variable exponent Lebesgue spaces, J. Funct. Anal. 257 (2009), no. 11, 3541–3551. 10.1016/j.jfa.2009.06.009Search in Google Scholar

[34] O. Kováčik and J. Rákosník, On spaces Lp(x)(Ω) and Wk,p(x)(Ω), Czechoslovak Math. J. 41 (1991), no. 4, 592–618. 10.21136/CMJ.1991.102493Search in Google Scholar

[35] P. Marcellini, Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations 90 (1991), no. 1, 1–30. 10.1016/0022-0396(91)90158-6Search in Google Scholar

[36] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), no. 1, 1–25. Search in Google Scholar

[37] M. Mihăilescu and V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2929–2937. 10.1090/S0002-9939-07-08815-6Search in Google Scholar

[38] M. Mihăilescu and V. Rădulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz–Sobolev spaces, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2087–2111. 10.5802/aif.2407Search in Google Scholar

[39] M. Mihăilescu, V. Rădulescu and D. Repovš, On a non-homogeneous eigenvalue problem involving a potential: An Orlicz–Sobolev space setting, J. Math. Pures Appl. (9) 93 (2010), no. 2, 132–148. 10.1016/j.matpur.2009.06.004Search in Google Scholar

[40] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 1983. 10.1007/BFb0072210Search in Google Scholar

[41] W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), no. 1, 200–211. 10.4064/sm-3-1-200-211Search in Google Scholar

[42] P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations 257 (2014), no. 5, 1529–1566. 10.1016/j.jde.2014.05.023Search in Google Scholar

[43] V. D. Rădulescu, Nonlinear elliptic equations with variable exponent: Old and new, Nonlinear Anal. 121 (2015), 336–369. 10.1016/j.na.2014.11.007Search in Google Scholar

[44] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2015. 10.1201/b18601Search in Google Scholar

[45] D. Repovš, Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.) 13 (2015), no. 6, 645–661. 10.1142/S0219530514500420Search in Google Scholar

[46] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000. 10.1007/BFb0104029Search in Google Scholar

[47] S. G. Samko, Density of C0(𝐑n) in the generalized Sobolev spaces Wm,p(x)(𝐑n), Dokl. Akad. Nauk 369 (1999), no. 4, 451–454. Search in Google Scholar

[48] X. Wang and J. Yao, Compact embeddings between variable exponent spaces with unbounded underlying domain, Nonlinear Anal. 70 (2009), no. 10, 3472–3482. 10.1016/j.na.2008.07.005Search in Google Scholar

[49] X. Wang, J. Yao and D. Liu, High energy solutions to p(x)-Laplace equations of Schrödinger type, Electron. J. Differential Equations 136 (2015), 1–17. Search in Google Scholar

[50] J. Yao, Solutions for Neumann boundary value problems involving p(x)-Laplace operators, Nonlinear Anal. 68 (2008), no. 5, 1271–1283. 10.1016/j.na.2006.12.020Search in Google Scholar

[51] J. Yao and X. Wang, 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 (2008), no. 4, 1445–1453. 10.1016/j.na.2007.06.044Search in Google Scholar

[52] L. Yin, J. Yao, Q. Zhang and C. Zhao, Multiplicity of strong solutions for a class of elliptic problems without the Ambrosetti–Rabinowitz condition in N, preprint (2016), https://arxiv.org/abs/1607.00581. Search in Google Scholar

[53] L. Yin, J. Yao, Q. H. Zhang and C. S. Zhao, Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth, Discrete Contin. Dyn. Syst. 37 (2017), no. 4, 2207–2226. 10.3934/dcds.2017095Search in Google Scholar

[54] N. Yoshida, Picone identities for half-linear elliptic operators with p(x)-Laplacians and applications to Sturmian comparison theory, Nonlinear Anal. 74 (2011), no. 16, 5631–5642. 10.1016/j.na.2011.05.048Search in Google Scholar

[55] Q. Zhang, A strong maximum principle for differential equations with nonstandard p(x)-growth conditions, J. Math. Anal. Appl. 312 (2005), no. 1, 24–32. 10.1016/j.jmaa.2005.03.013Search in Google Scholar

[56] Q. H. Zhang and V. D. Rădulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. (9), to appear. 10.1016/j.matpur.2018.06.015Search in Google Scholar

[57] J. F. Zhao, Structure Theory of Banach Spaces (in Chinese), Wuhan University, Wuhan, 1991. Search in Google Scholar

[58] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 29 (1987), no. 1, 33–36. 10.1070/IM1987v029n01ABEH000958Search in Google Scholar

[59] C. K. Zhong, X. L. Fan and W. Y. Chen, Introduction to Nonlinear Functional Analysis, Lanzhou University, Lanzhou, 1998. Search in Google Scholar

Received: 2018-01-16
Accepted: 2018-03-07
Published Online: 2018-06-16
Published in Print: 2020-10-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 27.9.2023 from https://www.degruyter.com/document/doi/10.1515/acv-2018-0003/pdf
Scroll to top button