Nonlinear elliptic equations with variable exponent: Old and new

In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type {-div A(x, del u) = lambda vertical bar u vertical bar(q(x)-2)u in Omega u=0 on partial derivative Omega, where Omega is a bounded or an exterior domain of R-N and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane-Emden equation and we focus on new phenomena which are due to the presence of variable exponents. (C) 2014 Elsevier Ltd. All rights reserved.