Existence results to a nonlinear p(x)-Laplace equation with degenerate coercivity and zero-order term: renormalized and entropy solutions

References

• Zhikov VV. On the density of smooth functions in Sobolev–Orlicz spaces. Zap. Nauchn. Sem. S-Peterburg. Otdel. Mat. Inst. Steklov (POMI). 2004;310. (Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34]):67–81, 226.
   Google Scholar
• Dal Maso G, Murat F, Orsina L, Prignet A. Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). 1999;28:741–808.
   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.
   Web of Science ®Google Scholar
• Kováčik O, Rákosník J. On spaces Lp(x) and Wk,p(x). Czechoslovak Math. J. 1991;41:592–618.
   Web of Science ®Google Scholar
• Diening L, Harjulehto P, Hästö P, Røužička M. Lebesgue and Sobolev spaces with variable exponents. Vol. 2017, Lecture notes in mathematics. Heidelberg: Springer; 2011.
   Google Scholar
• Ružička M. Electrorheological fluids: modeling and mathematical theory. Vol. 1748, Lecture notes in mathematics. Berlin: Springer-Verlag; 2000.
   Google Scholar
• Sanchón M, Urbano JM. Entropy solutions for the p(x)-Laplace equation. Trans. Amer. Math. Soc. 2009;361:6387–6405.
   Web of Science ®Google Scholar
• Bénilan P, Boccardo L, Gallouët T, Gariepy R, Pierre M, Vázquez JL. An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). 1995;22:241–273.
   Google Scholar
• Bendahmane M, Wittbold P, Zimmermann A. Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1-data. J. Differ. Equ. 2010;249:1483–1515.
   Web of Science ®Google Scholar
• Boccardo L, Giachetti D, Diaz JI, Murat F. Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differ. Equ. 1993;106:215–237.
   Web of Science ®Google Scholar
• Boccardo L, Croce G, Orsina L. Existence of solutions for some noncoercive elliptic problems involving derivatives of nonlinear terms. Differ. Equ. Appl. 2012;4:3–9.
   Google Scholar
• DiPerna RJ, Lions PL. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. (2). 1989;130:321–366.
   Web of Science ®Google Scholar
• Stampacchia G. Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus [The Dirichlet problem for the second order elliptic equations with discontinuous coefficients]. Ann. Inst. Fourier (Grenoble). 1965;15:189–258.
   Google Scholar
• Zeidler E. Nonlinear functional analysis and its applications. II, B. New York (NY): Springer-Verlag; 1990; Nonlinear monotone operators. Translated from the German by the author and Leo F.
   Google Scholar
• Leone C, Porretta A. Entropy solutions for nonlinear elliptic equations in L1. Nonlinear Anal. 1998;32:325–334.
   Web of Science ®Google Scholar
• Alvino A, Boccardo L, Ferone V, Orsina L, Trombetti G. Existence results for nonlinear elliptic equations with degenerate coercivity. Ann. Mat. Pura Appl. (4). 2003;182:53–79.
   Google Scholar
• Croce G. The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity. Rend. Mat. Appl. (7). 2007;27:299–314.
   Google Scholar
• Boccardo L, Dall’Aglio A, Orsina L. Existence and regularity results for some elliptic equations with degenerate coercivity. Atti. Sem. Mat. Fis. Univ. Modena. 1998;46:51–81; dedicated to Prof. C. Vinti (Italian) (Perugia, 1996).
   Google Scholar
• Bendahmane M, Wittbold P. Renormalized solutions for nonlinear elliptic equations with variable exponents and L1 data. Nonlinear Anal. 2009;70:567–583.
   Web of Science ®Google Scholar
• Ouaro S, Soma S. Weak and entropy solutions to nonlinear Neumann boundary-value problems with variable exponents. Complex Var. Elliptic Equ. 2011;56:829–851.
   Web of Science ®Google Scholar
• Zhang C. Entropy solutions for nonlinear elliptic equations with variable exponents. Elect. J. Differ. Equ. 2014;92:1–14.
   Google Scholar
• Zhang C, Zhou S. Entropy and renormalized solutions for the p(x)-Laplacian equation with measure data. Bull. Aust. Math. Soc. 2010;82:459–479.
   Web of Science ®Google Scholar
• Boccardo L, Gallouët T. Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 1989;87:149–169.
   Web of Science ®Google Scholar
• Lions JL. Quelques méthodes de résolution des problèmes aux limites non linéaires [Some methods for the nonlinear resolution limit problems]. Dunod; 1969.
   Google Scholar
• Orsina L, Prignet A. Non-existence of solutions for some nonlinear elliptic equations involving measures. Proc. Roy. Soc. Edinburgh Sect. A. 2000;130:167–187.
   Google Scholar
• DiBenedetto E. Real analysis. Birkhäuser advanced texts: Basler Lehrbücher. [Birkhäuser advanced texts: Basel Textbooks]. Boston (MA): Birkhäuser Boston; 2002.
   Google Scholar
• Blanchard D. Truncations and monotonicity methods for parabolic equations. Nonlinear Anal. 1993;21:725–743.
   Web of Science ®Google Scholar
• Blanchard D, Murat F, Redwane H. Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems. J. Differ. Equ. 2001;177:331–374.
   Web of Science ®Google Scholar
• Guibé O, Mercaldo A. Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data. Trans. Amer. Math. Soc. 2008;360:643–669 (electronic).
   Web of Science ®Google Scholar
• Murat F, Porretta A. Stability properties, existence, and nonexistence of renormalized solutions for elliptic equations with measure data. Commun. Par. Differ. Equ. 2002;27:2267–2310.
   Web of Science ®Google Scholar