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