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

Abstract

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