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 , D v ) d x under p,q p , q -growth conditions of the type λ(x)|ξ| p ≤f(x,ξ)≤μ(x)(1+|ξ| q ) λ ( x ) | ξ | p ≤ f ( x , ξ ) ≤ μ ( x ) 1 + | ξ | q for some exponents q≥p>1 q ≥ p > 1 and with nonnegative functions λ,μ λ , μ satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [ 26 ], where λ(x) λ ( x ) and μ(x) μ ( x ) had the role of the minimum and the maximum eigenvalues of an n×n n × n symmetric matrix (a ij (x)) a i j x and f ( x , ξ ) = ∑ i , j = 1 n a i j x ξ i ξ j 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) f ( x , ξ ) satisfying the general growth conditions above.