Weak and entropy solutions to nonlinear Neumann boundary-value problems with variable exponents

References

• Chen , Y , Levine , S and Rao , M . 2006 . Variable exponent, linear growth functionals in image restoration . SIAM. J. Appl. Math. , 66 : 1383 – 1406 .
   Google Scholar
• L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, University of Freiburg, Germany, 2002.
   Google Scholar
• Rajagopal , KR and Ruzicka , M . 2001 . Mathematical modeling of electrorheological materials . Contin. Mech. Thermodyn. , 13 : 59 – 78 .
   Google Scholar
• Ruzicka , M . 2002 . Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Mathematics , Vol. 1748 , Berlin : Springer-Verlag .
   Google Scholar
• Boureanu , M and Mihailescu , M . 2008 . Existence and multiplicity of solutions for a Neumann problem involving variable exponent growth conditions . Glasg. Math. J. , 50 : 565 – 574 .
   Google Scholar
• Yao , J . 2008 . Solutions for Neumann boundary value problems involving p(x)-Laplace operators . Nonlinear Anal. , 68 : 1271 – 1283 .
   Google Scholar
• Wang , L , Fan , Y and Ge , W . 2009 . Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator . Nonlinear Anal. , 71 : 4259 – 4270 .
   Google Scholar
• Ricceri , B . 2000 . On three critical points theorem . Arch. Math. (Basel) , 75 : 220 – 226 .
   Google Scholar
• Sanchon , M and Urbano , JM . 2009 . Entropy solutions for the p(x)-Laplace equation . Trans. Amer. Math. Soc. , 361 ( 12 ) : 6387 – 6405 .
   Google Scholar
• Bendahmane , M and Wittbold , P . 2009 . Renormalized solutions for nonlinear elliptic equations with variable exponents and L 1-data . Nonlinear Anal. Theory Methods Appl. , 70 ( 2 (A) : 567 – 583 .
   Google Scholar
• Ouaro , S and Bonzi , BK . 2010 . Entropy solutions for a doubly nonlinear elliptic problem with variable exponent . J. Math. Anal. Appl. , 370 ( 2 ) : 392 – 405 .
   Google Scholar
• Ouaro , S and Traoré , S . 2009 . Existence and uniqueness of entropy solutions to nonlinear elliptic problems with variable growth . Int. J. Evol. Equ. , 4 ( 4 ) : 89 – 109 .
   Google Scholar
• Ouaro , S and Traoré , S . 2009 . Weak and entropy solutions to nonlinear elliptic problems with variable exponent . J. Convex Anal. , 16 ( 2 ) : 523 – 541 .
   Google Scholar
• Wittbold , P and Zimmermann , A . 2010 . Existence and uniqueness of renormalized solutions to nonlinear elliptic equations with variable exponent and L 1-data . Nonlinear Anal. Theory Methods Appl. , 72 : 2990 – 3008 .
   Google Scholar
• Zhang , C and Zhou , S . 2010 . Renormalized and entropy solutions for nonlinear parabolic equations with variable exponent and L 1-data . J. Differ. Equ. , 248 ( 6 ) : 1376 – 1400 .
   Google Scholar
• Le , VK . 2009 . On sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces . Nonlinear Anal. , 71 : 3305 – 3321 .
   Google Scholar
• Kovacik , O and Rakosnik , J . 1991 . On spaces L p(x) and W 1,p(x) . Czech. Math. J. , 41 : 592 – 618 .
   Google Scholar
• Fan , X and Zhao , D . 2001 . On the spaces L p(x)(Ω) and W m,p(x)(Ω) . J. Math. Anal. Appl. , 263 : 424 – 446 .
   Google Scholar
• Fan , X and Zhang , Q . 2003 . Existence of solutions for p(x)-Laplacian Dirichlet problem . Nonlinear Anal. , 52 : 1843 – 1852 .
   Google Scholar
• Bénilan , P , Boccardo , L , Gallouèt , T , Gariepy , R , Pierre , M and Vazquez , JL . 1995 . An L 1 theory of existence and uniqueness of nonlinear elliptic equations . Ann Sc. Norm. Super. Pisa , 22 ( 2 ) : 240 – 273 .
   Google Scholar
• Andreu , F , Igbida , N , Mazón , JM and Toledo , J . 2007 . L 1 existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions . Ann. Inst. H. Poincaré Anal. Non Linéaire , 24 : 61 – 89 .
   Google Scholar
• Andreu , F , Mazón , JM , Segura De Léon , S and Toledo , J . 1997 . Quasi-linear elliptic and parabolic equations in L 1 with nonlinear boundary conditions . Adv. Math. Sci. Appl. , 7 ( 1 ) : 183 – 213 .
   Google Scholar
• Halmos , P . 1950 . Measure Theory , New York : D. Van Nostrand .
   Google Scholar
• Alvino , A , Boccardo , L , Ferone , V , Orsina , L and Trombetti , G . 2003 . Existence results for non-linear elliptic equations with degenerate coercivity . Ann. Mat. Pura Appl. , 182 : 53 – 79 .
   Google Scholar
• Antontsev , SN and Rodrigues , JF . 2006 . On stationary thermo-rheological viscous flows . Ann. Univ. Ferrara , 52 : 19 – 36 .
   Google Scholar
• Brezis , H . 1983 . Analyse Fonctionnelle: Théorie et Applications , Paris : Masson .
   Google Scholar
• Diening , L . 2004 . Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(·) and W 1,p(·) . Math. Nachr. , 268 : 31 – 43 .
   Google Scholar
• Edmunds , DE and Rakosnik , J . 1992 . Density of smooth functions in W k,p(x)(Ω) . Proc. R. Soc. A , 437 : 229 – 236 .
   Google Scholar
• Edmunds , DE and Rakosnik , J . 2000 . Sobolev embeddings with variable exponent . Stud. Math. , 143 : 267 – 293 .
   Google Scholar
• Edmunds , DE and Rakosnik , J . 2002 . Sobolev embeddings with variable exponent: II . Math. Nachr. , 246-247 : 53 – 67 .
   Google Scholar
• Ghergu , M and Radulescu , V . 2008 . Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications , Vol. 37 , Oxford, , UK : Oxford University Press .
   Google Scholar
• Krasnosel'skii , M . 1964 . Topological Methods in the Theory of Nonlinear Integral Equations , New York : Pergamon Press .
   Google Scholar
• Kristaly , A , Radulescu , V and Varga , C . 2010 . Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications , Vol. 136 , Cambridge : Cambridge University Press .
   Google Scholar
• Mihailescu , M and Radulescu , V . 2006 . A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids . Proc. R. Soc. A , 462 : 2625 – 2641 .
   Google Scholar
• Mihailescu , M and Radulescu , V . 2007 . On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent . Proc. Amer. Math. Soc. , 135 : 2929 – 2937 .
   Google Scholar
• Musielak , J . 1983 . Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics , Vol. 1034 , Berlin : Springer .
   Google Scholar
• Nakano , H . 1950 . Modulated Semi-ordered Linear Spaces , Tokyo : Maruzent .
   Google Scholar
• Sharapudinov , I . 1978 . On the topology of the space L p(t)([0,1]) . Mat. Zametki , 26 : 613 – 632 .
   Google Scholar
• Tsenov , I . 1961 . Generalization of the problem of best approximation of a function in the space L s . Uch. Zap. Dagestan Gos. Univ. , 7 : 25 – 37 .
   Google Scholar
• Zhikov , V . 1992 . On passing to the limit in nonlinear variational problem . Math. Sb. , 183 : 47 – 84 .
   Google Scholar