Nonuniformly elliptic energy integrals with p, q-growth

We study the local boundedness of minimizers of a nonuniformly energy integral of the form integral(Omega) f (x, Dv) dx under p, q-growth conditions of the type lambda(x)vertical bar xi vertical bar(p) <= f(x, xi) <= mu(x) (1+vertical bar xi vertical bar(q)) for some exponents q >= p > 1 and with nonnegative functions lambda,mu it satisfying some summability conditions. We use here the original notation introduced in 1971 by Trudinger [26], where lambda(x) and mu(x) had the role of the minimum and the maximum eigenvalues of an n x n symmetric matrix (a(ij) (x)) and f (x, xi) = Sigma(n)(i,j=1) a(ij) (x) xi(i)xi(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, xi) satisfying the general growth conditions above. (C) 2018 Elsevier Ltd. All rights reserved.