Nonuniformly elliptic energy integrals with p,q-growth

We study the local boundedness of minimizers of a nonuniformly energy integral of the form ∫ Ω f(x,Dv) dx under p, q-growth conditions of the type λ(x)|ξ|p ≤ f(x, ξ) ≤ μ(x) (1 + |ξ|q) for some exponents q ≥ p > 1 and with nonnegative functions λ, μ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [26], where λ(x) and μ(x) had the role of the minimum and the maximum eigenvalues of an n × n symmetric matrix (aij (x)) and f(x, ξ) =Σni,j=1 aij (x) ξiξj was the energy integrand associated to a linear nonuniformly elliptic equation in divergence form. In this paper we consider a class of energy integrals, associated to nonlinear nonuniformly elliptic equations and systems, with integrands f(x, ξ) satisfying the general growth conditions above.