We establish the local boundedness of the local minimizers $u:\Omega\rightarrow\mathbb{ R}^{ m}$ of non-uniformly elliptic integrals of the form $\int_{ \Omega}f(x,Dv)\,dx$, where $\Omega$ is a bounded open subset of $\mathbb{ R}^{ n}$ ($n\geq2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{ i=1}^{ n}\lambda_{ i}(x)|\xi_{ i}|^{ p_{ i}}\le f(x,\xi)\le\mu(x)\left\{ 1+|\xi|^{ q}\right\} \] for some exponents $q\geq p_{ i}>1$ and with non-negative functions $\lambda_{ i},\mu$ fulfilling suitable summability assumptions. The main novelties here are the degenerate and anisotropic behaviour of the integrand and the fact that we also address the case of vectorial minimizers ($m>1$). Our proof is based on the celebrated Moser iteration technique and employs an embedding result for anisotropic Sobolev spaces.
Categories: math.AP