Let $f\in L^m(\Omega)$ for some $m>1$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ ($N\geq 2$). Consider the equation, $$ \Delta_p u=f(x) $$for $p=N$, then $u$ is bounded in $\Omega$. Moreover, $u$ is continuous upto the boundary. Can anyone help me with the solution of this one. I have got this question while going through the paper : Lemma 3.7 of the paper below https://link.springer.com/content/pdf/10.1007%2Fs00030-016-0361-6.pdf
Thank you very much.
I'm not sure what your assumptions are, but I assume $p$ is equal to the dimension of the domain.
In the special case $f \equiv 0$ the continuity of $u$ can be easily shown. Look up Peter Lindqvist's Notes on the $p$-Laplace equation, section 3.2. The proof can be adjusted to work in the case of sufficiently regular nonzero right-hand side $f$.
As for the continuity-up-to-the boundary, this obviously depends on the boundary data. If one solves the equation with non-continuous boundary data (chosen in the trace space for $W^{1,n}$), then the solution is not in $C(\overline{\Omega})$.