On the continuity of solutions to the anisotropic $N$-Laplacian with $L^1$ lower order term

We establish the continuity of bounded solutions to the anisotropic elliptic equation $$-\sum _{i=1}^N(\mathrm{\mid }{u}_{x_i}{\mathrm{\mid }}^{p_i-2}{u}_{x_i}{)}_{x_i}=f(x),x\in \mathrm{\Omega },f(x)\in L^1(\mathrm{\Omega })$$ under the conditions $$\underset{1⩽i⩽N}{\min }{p}_i>1,\sum _{i=1}^N\frac{1}{p_i}=1$$ and $$\underset{\rho \to 0}{\lim }\underset{x\in \mathrm{\Omega }}{\sup }\underset{0}{\overset{\rho }{\int }}(\underset{B_r(x)}{\int }\mathrm{\mid }f(y)\mathrm{\mid }dy{)}^{\frac{1}{N-1}}\frac{dr}{r}=0.$$ In the standard case $p_1=\mathrm{.}\mathrm{.}\mathrm{.}=p_N=N$, these conditions recover the known results for the $N$-Laplacian.