Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights

We study the local boundedness of minimizers of non uniformly elliptic integral functionals with a suitable anisotropic p,q- growth condition. More precisely, the growth condition of the integrand function f(x,∇u) from below involves different p>1 powers of the partial derivatives of u and some monomial weights |x| with α∈[0,1) that may degenerate to zero. Otherwise from above it is controlled by a q power of the modulus of the gradient of u with q≥maxp and an unbounded weight μ(x). The main tool in the proof is an anisotropic Sobolev inequality with respect to the weights |x|.