Boundedness of solutions of the non-uniformly convex, non-standard growth Laplacian

References

• Diening , L , Hästö , P and Nekvinda , A . 2004 . “ Open problems in variable exponent Lebesgue and Sobolev spaces ” . In FSDONA04 Proceedings , Edited by: Drabek , P and Rakosnik , J . 38 – 58 . Czech Republic : Milovy .
   Google Scholar
• Harjulehto , P , Hästö , P , Lê , Ú and Nuortio , M . 2009 . Overview of differential equations with non-standard growth, preprint , Nonlinear Anal., to appear
   Google Scholar
• Samko , S . 2005 . On a progress in the theory of Lebesgue spaces with variable exponent: Maximal and singular operators . Integral Transforms Spec. Funct. , 16 ( 5–6 ) : 461 – 482 .
   Google Scholar
• Diening , L , Harjulehto , P , Hästö , P and Růžička , M . 2010 . Lebesgue and Sobolev Spaces with Variable Exponents (submitted)
   Google Scholar
• Cruz-Uribe , D , Fiorenza , A , Martell , JM and Pérez , C . 2006 . The boundedness of classical operators on variable L p spaces . Ann. Acad. Sci. Fenn. Math. , 31 ( 1 ) : 239 – 264 .
   Google Scholar
• Edmunds , DE , Lang , J and Nekvinda , A . 2009 . Some s-numbers of an integral operator of Hardy type on L p(·) spaces . J. Funct. Anal. , 257 ( 1 ) : 219 – 242 .
   Google Scholar
• Lerner , AK . 2009 . On some questions related to the maximal operator on variable L p spaces . Trans. Amer. Math. Soc., preprint , Nonlinear Anal., to appear
   Google Scholar
• Mizuta , Y , Ohno , T and Shimomura , T . 2008 . Sobolev's inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space L p(·)(log L) q(·) . J. Math. Anal. Appl. , 345 ( 1 ) : 70 – 85 .
   Google Scholar
• Kokilashvili , V and Samko , S . 2009 . Operators of harmonic analysis in weighted spaces with non-standard growth . J. Math. Anal. Appl. , 352 ( 1 ) : 15 – 34 .
   Google Scholar
• Acerbi , E and Mingione , G . 2002 . Regularity results for stationary electro-rheological fluids . Arch. Ration. Mech. Anal. , 164 ( 3 ) : 213 – 259 .
   Google Scholar
• Růžička , M . 2000 . “ Electrorheological Fluids: Modeling and Mathematical Theory ” . In Lecture Notes in Mathematics, Vol. 1748 , Berlin : Springer-Verlag .
   Google Scholar
• Chen , Y , Levine , S and Rao , M . 2006 . Variable exponent, linear growth functionals in image restoration . SIAM J. Appl. Math. , 66 ( 4 ) : 1383 – 1406 .
   Google Scholar
• Acerbi , E and Mingione , G . 2001 . Regularity results for a class of functionals with non-standard growth . Arch. Ration. Mech. Anal. , 156 : 121 – 140 .
   Google Scholar
• Antontsev , S , Chipot , M and Xie , Y . 2007 . Uniqueness results for equations of the p(x)-Laplacian type . Adv. Math. Sci. Appl. , 17 ( 1 ) : 287 – 304 .
   Google Scholar
• Coscia , A and Mingione , G . 1999 . Hölder continuity of the gradient of p(x)-harmonic mappings . C. R. Acad. Sci. Paris , 328 : 363 – 368 .
   Google Scholar
• Dai , G and Liu , D . 2009 . Infinitely many positive solutions for a p(x)-Kirchhoff-type equation . J. Math. Anal. Appl. , 359 ( 2 ) : 704 – 710 .
   Google Scholar
• Fan , X-L . 2007 . Global C 1,α regularity for variable exponent elliptic equations in divergence form . J. Diff. Eqns. , 235 ( 2 ) : 397 – 417 .
   Google Scholar
• Habermann , J and Zatorska-Goldstein , A . 2008 . Regularity for minimizers of functionals with nonstandard growth by 𝒜-harmonic approximation . NoDEA Nonlinear Diff. Eqns. Appl. , 15 ( 1–2 ) : 169 – 194 .
   Google Scholar
• Sanchón , M and Urbano , J . 2009 . Entropy solutions for the p(x)-Laplace equation . Trans. Amer. Math. Soc. , 361 ( 12 ) : 6387 – 6405 .
   Google Scholar
• Yao , J-H . 2008 . Solutions for Neumann boundary value problems involving p(x)-Laplace operators . Nonlinear Anal. , 68 ( 5 ) : 1271 – 1283 .
   Google Scholar
• Zhikov , VV . 1997 . On some variational problems . Russial J. Math. Physics , 5 : 105 – 116 .
   Google Scholar
• Harjulehto , P , Hästö , P and Latvala , V . 2008 . Minimizers of the variable exponent, non-uniformly convex Dirichlet energy . J. Math. Pures Appl. (9) , 89 ( 2 ) : 174 – 197 .
   Google Scholar
• Harjulehto , P , Hästö , P and Latvala , V . 2009 . Harnack's inequality for p(·)-harmonic functions with unbounded exponent p . J. Math. Anal. Appl. , 352 ( 1 ) : 345 – 359 .
   Google Scholar
• Lindqvist , P and Lukkari , T . 2009 . A curious equation involving the infinity-Laplacian . Adv. Calc. Var., preprint , Nonlinear Anal., to appear
   Google Scholar
• Manfredi , JJ , Rossi , JD and Urbano , JM . 2009 . p(x)-harmonic function with unbounded exponent in a subdomain . Ann. Inst. H. Poincaré Anal. Non Linéaire , 26 ( 6 ) : 2581 – 2595 .
   Google Scholar
• Manfredi , JJ , Rossi , JD and Urbano , JM . 2010 . Limits as p(x) → ∞ of p(x)-harmonic functions . Nonlinear Anal. , 72 : 309 – 315 .
   Google Scholar
• Adamowicz , T and Hästö , P . Mappings of finite distortion and p(·)-harmonic functions . Int. Math. Res. Not. IMRN , doi: 1093/imrn/rnp192, preprint (2009) Nonlinear Anal., to appear
   Google Scholar
• Alkhutov , Yu . 1997 . The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition . Diff. Eqns. , 33 ( 12 ) : 1653 – 1663 .
   Google Scholar
• Serrin , J . 1964 . Local behaviour of solutions of quasi-linear equations . Acta Math. , 111 : 247 – 302 .
   Google Scholar
• Kováčik , O and Rákosník , J . 1991 . On spaces L p(x) and W 1,p(x) . Czechoslovak Math. J. , 41 ( 116 ) : 592 – 618 .
   Google Scholar
• Zhikov , VV . 1995 . On Lavrentiev's phenomenon . Russial J. Math. Physics , 3 : 249 – 269 .
   Google Scholar
• Fan , X-L . 1996 . Regularity of nonstandard Lagrangians f(x, ξ) . Nonlinear Anal. , 27 ( 6 ) : 669 – 678 .
   Google Scholar
• Diening , L . 2004 . Maximal function on generalized Lebesgue spaces L p(·) . Math. Inequal. Appl. , 7 ( 2 ) : 245 – 253 .
   Google Scholar
• Zhikov , VV . 2006 . On the density of smooth functions in Sobolev-Orlicz spaces . J. Math. Sci. (N. Y.) , 132 ( 3 ) : 285 – 294 .
   Google Scholar
• Evans , LC and Gariepy , RF . 1992 . Measure Theory and Fine Properties of Functions , Boca Raton : CRC Press .
   Google Scholar
• Hästö , P . 2006 . On the variable exponent Dirichlet energy integral . Comm. Pure Appl. Anal. , 5 ( 3 ) : 413 – 420 .
   Google Scholar
• Harjulehto , P , Hästö , P , Koskenoja , M and Varonen , S . 2006 . The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values . Potential Anal. , 25 ( 3 ) : 205 – 222 .
   Google Scholar
• Malý , J and Ziemer , WP . 1997 . Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs, Vol. 51 , Providence, RI : American Mathematical Society .
   Google Scholar
• Harjulehto , P , Hästö , P and Koskenoja , M . 2007 . Properties of capacity in variable exponent Sobolev spaces . J. Anal. Appl. , 5 ( 2 ) : 71 – 92 .
   Google Scholar
• Harjulehto , P , Kinnunen , J and Lukkari , T . 2007 . Unbounded supersolutions of nonlinear equations with nonstandard growth . Bound. Value Probl. , Article ID 48348, 20 pages, doi:10.1155/2007/48348. preprint (2009) Nonlinear Anal., to appear
   Google Scholar