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

Abstract

Abstract We establish the continuity of bounded solutions to the anisotropic elliptic equation $$\begin{aligned} -\sum \limits _{i=1}^N\Big (|u_{x_i}|^{p_i-2} u_{x_i}\Big )_{x_i}=f(x),\quad x\in \Omega ,\quad f(x)\in L^1(\Omega ) \end{aligned}$$ under the conditions $$\begin{aligned} \min \limits _{1\leqslant i\leqslant N} p_i >1,\quad \sum \limits _{i=1}^N \frac{1}{p_i}=1 \end{aligned}$$ and $$\begin{aligned} \lim \limits _{\rho \rightarrow 0}\,\sup \limits _{x\in \Omega }\int \limits ^{\rho }_0\Big (\int \limits _{B_r(x)}|f(y)|\,dy\Big )^{\frac{1}{N-1}}\frac{dr}{r}=0. \end{aligned}$$ In the standard case $$p_1=...=p_N=N$$ , these conditions recover the known results for the N -Laplacian.