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 \(f(x,\nabla u)\) from below involves different \(p_i>1\) powers of the partial derivatives of u and some monomial weights \(|x_i|^{\alpha _i p_i}\) with \(\alpha _i \in [0,1)\) that may degenerate to zero. Otherwise from above it is controlled by a q power of the modulus of the gradient of u with \(q\ge \max _i p_i\) and an unbounded weight \(\mu (x)\). The main tool in the proof is an anisotropic Sobolev inequality with respect to the weights \(|x_i|^{\alpha _i p_i}\).