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 , ∇ 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 | α i p i with $$\alpha _i \in [0,1)$$ α i ∈ [ 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 ≥ max i p i and an unbounded weight $$\mu (x)$$ μ ( 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 | α i p i .