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{−divA(x,∇u)=λ|u|q(x)−2uinΩu=0on∂Ω, where ΩΩ is a bounded or an exterior domain of RNRN and qq 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.