Existence of solutions for nonlinear elliptic variational inequalities with L¹ data

References

• Eleuteri M, Habermann J. Regularity results for a class of obstacle problems under nonstandard growth conditions. J Math Anal Appl. 2008;344:1120–1142. doi: 10.1016/j.jmaa.2008.03.068
   Web of Science ®Google Scholar
• Harjulehto P, Hästö P, Koskenoja M, et al. An obstacle problem and superharmonic functions with nonstandard growth. Nonlinear Anal TMA. 2007;67(12):3424–3440. doi: 10.1016/j.na.2006.10.026
   Web of Science ®Google Scholar
• Ouaro S, Traore S. Entropy solutions to the obstacle problem for nonlinear elliptic problems with variable exponent and L1-data. Pac J Optim. 2009;5(1):127–141.
   Web of Science ®Google Scholar
• Rodrigues JF, Sanchón M, Urbano JM. The obstacle problem for nonlinear elliptic equations with variable growth and L1-data. Monatsh Math. 2008;154:303–322. doi: 10.1007/s00605-008-0550-4
   Web of Science ®Google Scholar
• Sanchón M, Urbano JM. Entropy solutions for the p(x)-Laplace equation. Trans Amer Math Soc. 2009;361:6387–6405. doi: 10.1090/S0002-9947-09-04399-2
   Web of Science ®Google Scholar
• Zhang C, Zhou SL. Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L1 data. J Differ Equ. 2010;248:1376–1400. doi: 10.1016/j.jde.2009.11.024
   Web of Science ®Google Scholar
• Zou WL, Wang W, Bi YH. On a class of nonlinear obstacle problems with nonstandard growth. Math Method Appl Sci. 2015;38:2911–2921. doi: 10.1002/mma.v38.14
   Web of Science ®Google Scholar
• Acerbi E, Mingione G, Seregin GA. Regularity results for parabolic systems related to a class of non-Newtonian fluids. Ann Inst H Poincaré Anal Non Linéaire. 2004;21(1):25–60. doi: 10.4171/aihpc
   Web of Science ®Google Scholar
• R˙užička M. Electrorheological fluids: modelling and mathematical theorey. Berlin: Springer; 2000. (Lecture Notes in Math; 1748).
   Google Scholar
• Chen YM, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math. 2006;66(4):1383–1406. doi: 10.1137/050624522
   Web of Science ®Google Scholar
• Boccardo L, Gallouët T. Problèmes unilatéraux avec données dans L1. C R Acad Sci Paris. 1990;311:617–619.
   Google Scholar
• Boccardo L, Cirmi R. Existence and uniqueness of solutions of unilateral problems with L1 data. J Convex Anal. 1999;6:195–206.
   Web of Science ®Google Scholar
• Rakotoson JM, Temam R. Relative rearrangement in quasilinear elliptic variational inequalities. Indiana Univ Math J. 1987;36:757–810. doi: 10.1512/iumj.1987.36.36043
   Web of Science ®Google Scholar
• Ferone V, Posteraro MR, Rakotoson JM. L∞-estimates for nonlinear elliptic problems with p-growth in the gradient. J Inequal Appl. 1999;3(2):109–125.
   Web of Science ®Google Scholar
• Arcoya D, Boccardo L. Regularizing effect of the interplay between coefficients in some elliptic equations. J Funct Anal. 2015;268(5):1053–1077. doi: 10.1016/j.jfa.2014.11.011
   Web of Science ®Google Scholar
• Arcoya D, Boccardo L. Regularizing effect of Lq interplay between coefficients in some elliptic equations. J Math Pures Appl. 2018;111(9):106–125. doi: 10.1016/j.matpur.2017.08.001
   Google Scholar
• Arcoya D, Boccardo L, Orsina L. Regularizing effect of the interplay between coefficients in some nonlinear Dirichlet problems with distributional data. Ann Mat Pura Appl. 2020;199:1909–1921. doi: 10.1007/s10231-020-00949-8
   Web of Science ®Google Scholar
• Li ZQ. Existence result to a parabolic equation with quadratic gradient term and an L1 source. Acta Appl Math. 2019;163(1):145–156. doi: 10.1007/s10440-018-0217-7
   Web of Science ®Google Scholar
• Moreno L, Porzio MM. Existence and asymptotic behavior of a parabolic equation with L1 data. Asymptotic Anal. 2020;118(3):143–159. doi: 10.3233/ASY-191558
   Web of Science ®Google Scholar
• Boccardo L, Murat F, Puel JP. Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann Mat Pura Appl. 1988;152(1):183–196. doi: 10.1007/BF01766148
   Google Scholar
• Diening L, Harjulehto P, Hästö P, et al. Lebesgue and Sobolev spaces with variable exponents. Berlin: Springer; 2011. (Lecture Notes in Math; 2017).
   Google Scholar
• Fan X, Zhao D. On the spaces Lp(x)(Ω) and Wm,p(x)(Ω). J Math Anal Appl. 2001;263:424–446. doi: 10.1006/jmaa.2000.7617
   Web of Science ®Google Scholar
• Lions JL. Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Paris: Dunod Gauthier-Villars; 1969.
   Google Scholar
• Stampacchia G. Le probleme de Dirichlet pour les equation elliptiques du second ordre a coefficients discontinus. Ann I Fourier. 1965;15:189–257. doi: 10.5802/aif.204
   Google Scholar