Existence of solutions to a class of elliptic equations with nonstandard growth condition and zero order term

Abstract

The existence of bounded weak solutions, to a class of nonlinear elliptic equations with variable exponents, is investigated in this article. A uniform a priori L^{\mathrm{\infty }} estimate is obtained by the De Giorgi iterative technique. Thanks to the weak convergence method and Minty's trick, the existence result is proved through limit process.

Keywords

Elliptic equations / variable exponents / De Giorgi iteration / Minty's trick

References

[1] AdamsR AFournierJ J F. Sobolev Spaces. Amsterdam: Elsevier/Academic Press, 2003

[2] Bendahmane M. Renormalized solutions for nonlinear elliptic equations with variable exponents and L1 data. Nonlinear Anal 2009; 70(2): 567–583

[3] Bendahmane M, Wittbold P. Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1-data. J Differential Equations 2010; 249(6): 1483–1515

[4] Betta M F, Guibé O, Mercaldo A. Neumann problems for nonlinear elliptic equations with L1 data. J Differential Equations 2015; 259(3): 898–924

[5] BoccardoLCroceG. Elliptic Partial Differential Equations: Existence and Regularity of Distributional Solutions. Berlin: De Gruyter, 2014

[6] Boccardo L, Marcellini P, Sbordone C. L∞-regularity for variational problems with sharp nonstandard growth conditions. Boll Un Mat Ital 1990; 4(2): 219–225

[7] Boccardo L. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal 1992; 19(6): 581–597

[8] Boccardo L, Murat F, Puel J P. L∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result. SIAM J Math Anal 1992; 23(2): 326–333

[9] Boccardo L, Orsina L. Leray-Lions operators with logarithmic growth. J Math Anal Appl 2015; 423(1): 608–622

[10] Bulíček M, Gwiazda P, Skrzeczkowski J. Parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function. J Differential Equations 2021; 290: 17–56

[11] Fan Xianling, Zhao Dun. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J Math Anal Appl 2001; 263(2): 424–446

[12] Kováčik O, Rákosník J R. On spaces Lp(x) and Wk,p(x). Czechoslovak Math J 1991; 41(4): 592–618

[13] LiZhongqingGaoWenjie. Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent. Bound Value Probl, 2014: Paper No 77, 21 pp

[14] Porzio M M. L∞-regularity for degenerate and singular anisotropic parabolic equations. Boll Un Mat Ital 1997; 11(3): 697–707

[15] RužičkaM. Electrorheological Fluids: Modeling and Mathematical Theory. Berlin: Springer-Verlag, 2000

[16] ZeidlerE. Nonlinear Functional Analysis and Its Applications, Ⅱ/B. New York: Springer-Verlag, 1990

[17] Zhang Chao, Zhou Shulin. Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L1 data. J Differential Equations 2010; 248(6): 1376–1400

RIGHTS & PERMISSIONS

Higher Education Press 2023