Overview of differential equations with non-standard growth

Abstract Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.

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[57]  Qihu Zhang.  Existence of positive solutions for elliptic systems with nonstandard p(x)-growth conditions via sub-supersolution method , 2007 .

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[60]  Xianling Fan,et al.  A Knobloch-type result for p(t)-Laplacian systems☆ , 2003 .

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[62]  Guowei Dai.  Infinitely many solutions for a hemivariational inequality involving the p(x)-Laplacian ☆ , 2009 .

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[65]  Mihai Mihailescu,et al.  A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[66]  J. Rossi,et al.  Existence, Asymptotic Behavior and Uniqueness for Large Solutions to Δu = eq(x)u , 2009 .

[67]  Jingjing Liu,et al.  Existence of three solutions for a class of quasilinear elliptic systems involving the (p(x), q(x))-Laplacian , 2009 .

[68]  Qihu Zhang.  Existence of solutions for p(x)-Laplacian equations with singular coefficients in RN☆ , 2008 .

[69]  V. Zhikov,et al.  Higher integrability for parabolic equations of $p(x,t)$-Laplacian type , 2005, Advances in Differential Equations.

[70]  Xianling Fan,et al.  Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , 2004 .

[72]  Xiaopin Liu,et al.  Existence of solutions and nonnegative solutions for weighted p(r)-Laplacian impulsive system multi-point boundary value problems☆ , 2009 .

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[77]  T. Lukkari,et al.  An obstacle problem and superharmonic functions with nonstandard growth , 2007 .

[78]  P. Hästö,et al.  The Dirichlet Energy Integral on Intervals in Variable Exponent Sobolev Spaces , 2003 .

[79]  Mihai Mihailescu.  Elliptic problems in variable exponent spaces , 2006, Bulletin of the Australian Mathematical Society.

[80]  John L. Lewis.  On very weak solutions of certain elliptic systems , 1993 .

[81]  T. Lukkari,et al.  A curious equation involving the ∞-Laplacian , 2009, 0902.1771.

[82]  T. Lukkari,et al.  Wolff potential estimates for elliptic equations with nonstandard growth and applications , 2010 .

[83]  Giuseppe Mingione,et al.  Regularity Results for a Class of Functionals with Non-Standard Growth , 2001 .

[84]  E. Acerbi,et al.  Regularity results for a class of quasiconvex functionals with nonstandard growth , 2001 .

[85]  Boundary Asymptotic and Uniqueness of Solution for a Problem with px -Laplacian , 2008 .

[86]  Peter Hästö,et al.  The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values , 2006 .

[87]  Driss Meskine,et al.  New diffusion models in image processing , 2008, Comput. Math. Appl..

[88]  Qihu Zhang.  Oscillatory Property of Solutions for-Laplacian Equations , 2007 .

[89]  Xianling Fan,et al.  Nodal solutions of p(x)-Laplacian equations , 2007 .

[90]  N. Wolanski,et al.  A free boundary problem for the p(x)-Laplacian , 2009, 0902.3216.

[91]  Vicentiu D. Rădulescu,et al.  Continuous spectrum for a class of nonhomogeneous differential operators , 2007, 0706.4045.

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[93]  L. Diening,et al.  Calderón-Zygmund operators on generalized Lebesgue spaces $L^{p(\cdot)}$ and problems related to fluid dynamics , 2003 .

[94]  Xianling Fan.  Eigenvalues of the p(x)-Laplacian Neumann problems☆ , 2007 .

[95]  Jinghua Yao.  Solutions for Neumann boundary value problems involving p(x)-Laplace operators , 2008 .

[96]  E. DiBenedetto.  Degenerate Parabolic Equations , 1993 .

[97]  Giuseppe Mingione,et al.  Gradient estimates for the p (x)-Laplacean system , 2005 .

[98]  Weigao Ge,et al.  Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator , 2007 .

[99]  M. Galewski.  On a Dirichlet problem with p(x)-Laplacian , 2008 .

[100]  Qihu Zhang,et al.  Existence of positive solutions for a class of p(x)-Laplacian systems ✩ , 2007 .

[101]  Peter Hästö,et al.  Harnack's inequality for p(⋅)-harmonic functions with unbounded exponent p , 2009 .

[102]  K. Kurata,et al.  Compact embedding from W01,2(Ω) to Lq(x)(Ω) and its application to nonlinear elliptic boundary value problem with variable critical exponent , 2008 .

[103]  T. Iwaniec,et al.  Weak minima of variational integrals. , 1994 .

[104]  D. Mucci,et al.  Integral representation and Γ-convergence of variational integrals with p(x)-growth , 2002 .

[105]  P. Hästö,et al.  Boundedness of solutions of the non-uniformly convex, non-standard growth Laplacian , 2011 .

[106]  Guowei Dai.  Infinitely many solutions for a p(x)-Laplacian equation in RN☆ , 2009 .

[107]  Yongqiang Fu,et al.  A multiplicity result for p(x)-Laplacian problem in RN , 2009 .

[108]  Xianling Fan,et al.  Global C1,α regularity for variable exponent elliptic equations in divergence form , 2007 .

[109]  John L. Lewis,et al.  Very weak solutions of parabolic systems ofp-Laplacian type , 2002 .

[110]  M. Eleuteri,et al.  A Hölder continuity result for a class of obstacle problems under non standard growth conditions , 2011 .

[111]  Erik M. Bollt,et al.  Graduated adaptive image denoising: local compromise between total variation and isotropic diffusion , 2009, Adv. Comput. Math..

[112]  On the C1,γ(Ω¯)∩W2,2(Ω) regularity for a class of electro-rheological fluids , 2009 .

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[114]  Shao-Gao Deng,et al.  A local mountain pass theorem and applications to a double perturbed p(x)-Laplacian equations , 2009, Appl. Math. Comput..

[115]  Xianling Fan.  p(x)-Laplacian equations in RN with periodic data and nonperiodic perturbations , 2008 .

[116]  M. Růžička.  Flow of Shear Dependent Electrorheological Fluids: Unsteady Space Periodic Case , 2002 .

[118]  Dun Zhao,et al.  The quasi-minimizer of integral functionals with m ( x ) growth conditions , 2000 .

[119]  Limits as p(x)→∞p(x)→∞ of p(x)p(x)-harmonic functions , 2010 .

[120]  Chao Ji.  Perturbation for a p(x)-Laplacian equation involving oscillating nonlinearities in RN☆ , 2008 .

[121]  T. Lukkari,et al.  Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth , 2006 .

[122]  Qihu Zhang.  Boundary Blow-Up Solutions to -Laplacian Equations with Exponential Nonlinearities , 2007 .

[123]  Qihu Zhang.  Existence and Asymptotic Behavior of Positive Solutions to-Laplacian Equations with Singular Nonlinearities , 2007 .

[124]  Xianling Fan,et al.  Existence and multiplicity of solutions for p(x)p(x)-Laplacian equations in RNRN☆ , 2004 .

[125]  P. Harjulehto,et al.  FINE TOPOLOGY OF VARIABLE EXPONENT ENERGY SUPERMINIMIZERS , 2008 .

[126]  M. Ruzicka,et al.  Electrorheological Fluids: Modeling and Mathematical Theory , 2000 .

[127]  Qihu Zhang,et al.  Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems , 2009 .

[128]  Xiaopin Liu,et al.  Existence of multiple solutions for weighted p(r)-Laplacian equation Dirichlet problems☆ , 2009 .

[129]  Mengrui Xu,et al.  Hölder Continuity of Weak Solutions for Parabolic Equations with Nonstandard Growth Conditions , 2006 .

[130]  J. Rodrigues,et al.  The obstacle problem for nonlinear elliptic equations with variable growth and L1-data , 2008, 0802.0378.

[131]  M. Eleuteri,et al.  Regularity results for a class of obstacle problems under nonstandard growth conditions , 2008 .

[132]  Giuseppina Autuori,et al.  Global Nonexistence for Nonlinear Kirchhoff Systems , 2010 .

[133]  Nikolaos S. Papageorgiou,et al.  A MULTIPLICITY THEOREM FOR A VARIABLE EXPONENT DIRICHLET PROBLEM , 2008, Glasgow Mathematical Journal.

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[135]  Higher integrability of very weak solutions of systems of p(x)-Laplacean type , 2007 .

[136]  P. Zhao,et al.  Existence of positive solutions for p(x)-Laplacian equations in unbounded domains , 2008 .

[137]  Qiao Liu.  Existence of three solutions for p(x)-Laplacian equations , 2008 .

[138]  Michel Ménard,et al.  Nonstandard diffusion in image restoration and decomposition , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

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[142]  J. Habermann.  Partial regularity for minima of higher order functionals with p(x)-growth , 2006, math/0610145.

[143]  Qihu Zhang.  Existence and asymptotic behavior of blow-up solutions to a class of p(x)-Laplacian problems☆ , 2007 .

[144]  Xianling Fan,et al.  Hartman-type results for p(t) -Laplacian systems , 2003 .

[145]  Xiaopin Liu,et al.  Existence of solutions for weighted p(r)-Laplacian impulsive system periodic-like boundary value problems☆ , 2009 .

[146]  M. Mihăilescu.  EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR AN ELLIPTIC EQUATION WITH $p(x)$-GROWTH CONDITIONS , 2006, Glasgow Mathematical Journal.

[147]  Vicentiu D. Rădulescu,et al.  Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent , 2008 .

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[149]  L. Diening,et al.  C1,α-regularity for electrorheological fluids in two dimensions , 2007 .

[150]  Xianling Fan,et al.  Remarks on Ricceri’s variational principle and applications to the p(x)-Laplacian equations , 2007 .

[151]  Giuseppe Mingione,et al.  Regularity Results for Stationary Electro-Rheological Fluids , 2002 .

[152]  U. Gianazza,et al.  Harnack estimates for quasi-linear degenerate parabolic differential equations , 2008 .

[153]  Qihu Zhang,et al.  Eigenvalues of p(x)-Laplacian Dirichlet problem , 2005 .

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[155]  R. Mashiyev,et al.  Existence of Solutions for a Class of Elliptic Systems in Involving the -Laplacian , 2008 .

[156]  Yukun An,et al.  Existence and multiplicity of solutions for elliptic systems with nonstandard growth condition in RN , 2008 .

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[158]  Qihu Zhang,et al.  On the boundary blow-up solutions of p(x)-Laplacian equations with singular coefficient , 2009 .

[159]  On the existence and stability of solutions for Dirichlet problem with p(x)-Laplacian , 2007 .

[160]  Dušan D. Repovš,et al.  On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting , 2016, 1608.07062.

[161]  Qihu Zhang,et al.  Existence of solutions for p(x) -Laplacian dirichlet problem , 2003 .

[162]  P. Pucci,et al.  Maximum principles for anisotropic elliptic inequalities , 2009 .

[163]  R. Yuan,et al.  Existence of periodic solutions for p(t)-Laplacian systems , 2009 .

[164]  J. M. Urbano,et al.  Intrinsic scaling for pde's with an exponential nonlinearity , 2006 .

[165]  C. O. Alves.  Existence of solution for a degenerate p(x)-Laplacian equation in RN , 2008 .

[166]  M. Bendahmane,et al.  Renormalized solutions for nonlinear elliptic equations with variable exponents and L1 data , 2009 .

[167]  A constrained minimization problem involving the p(x)-Laplacian in RN , 2008 .

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