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 $$\sum _{i=1}^n{\lambda }_i(x)\mathrm{\mid }{\xi }_i{\mathrm{\mid }}^{p_i}\le f(x,u,\xi )\le \mu (x)\{\mathrm{\mid }\xi {\mathrm{\mid }}^q+\mathrm{\mid }u{\mathrm{\mid }}^{\gamma }+1\}$$ 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.