Nonlinear Analysis 72 (2010) 4551–4574 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Overview of differential equations with non-standard growth Petteri Harjulehto a , Peter Hästö b , Út V. Lê b , Matti Nuortio b,∗ a Department of Mathematics and Statistics, P.O. Box 68, FI-00014, University of Helsinki, Finland b Department of Mathematical Sciences, P.O. Box 3000, FI-90014, University of Oulu, Finland article info Article history: Received 13 May 2009 Accepted 17 February 2010 MSC: 35J60 35J20 46E35 Keywords: Variable exponent Non-standard growth Eigenvalue problem Existence Uniqueness Regularity Harmonic functions Elliptic equations Parabolic equations 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. © 2010 Elsevier Ltd. All rights reserved. I. Introduction A detailed introduction to this article is given in this section. 1. Overview of the history and the article Differential equations with non-standard growth and corresponding function spaces with variable exponents have been a very active field of investigation in recent years. We counted over 300 publications from more than 100 authors this decade. The theory of variable exponent Lebesgue and Sobolev spaces has been surveyed in [1,2]; see also the upcoming monograph [3]. Mingione [4] has written an extensive exposition of regularity theory, including the non-standard growth case; Fan [5] summarized some results of his research group on the existence and multiplicity of solutions of eigenvalue problems. However, there has so far not been any comprehensive survey of differential equations with non-standard growth including and comparing results on existence and regularity. It is the purpose of the present article to fill this gap. Although we were obviously forced to make choices about what to include in detail, we have tried to include in the bibliography all works published to date (in international forums), with an indication in the text of where they fit in. ∗ Corresponding author. Fax: +358 8 553 1730. E-mail addresses: petteri.harjulehto@helsinki.fi (P. Harjulehto), peter.hasto@helsinki.fi (P. Hästö), levanut@gmail.com, ut.van.le@oulu.fi (U.V. Lê), mnuortio@paju.oulu.fi (M. Nuortio). 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.02.033