Local boundedness for solutions of a class of non-uniformly elliptic anisotropic problems

We consider a class of energy integrals, associated to nonlinear and non-uniformly elliptic equations, with integrands $f(x,u,\xi )$ satisfying anisotropic $p_i,q$-growth conditions of the form ${\sum }_{i=1}^n{\lambda }_i\left(x\right){\left|{\xi }_i\right|}^{p_i}\le f(x,u,\xi )\le \mu \left(x\right)\left({\left|\xi \right|}^q+{\left|u\right|}^{\gamma }+1\right)$ for some exponents $\gamma \ge q\ge p_i>1$, and non-negative functions ${\lambda }_i,\mu$ subject to suitable summability assumptions. We prove the local boundedness of scalar local quasi-minimizers of such integrals.