Nonuniformly elliptic energy integrals with [formula omitted]-growth

Abstract

We study the local boundedness of minimizers of a nonuniformly energy integral of the form ∫Ωf(x,Dv)dx under p,q-growth conditions of the type λ(x)|ξ|p≤f(x,ξ)≤μ(x)1+|ξ|qfor 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 aijx and f(x,ξ)=∑i,j=1naijxξiξjwas 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.