Consider $\Omega \subset R^n$ a bounded domain. Let $\varphi \in W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$. Let $u \in W^{1,p}(\Omega)$ with $\Delta_p u = 0$ in $\Omega$ with $u - \varphi\in W^{1,p}_{0}(\Omega)$ . Can I conclude that $u \in L^{\infty}(\Omega)$?
Intuitively the answer is "yes" (I believe that exists a comparison principle for the above situation). But I am not finding a reference. Someone could point me a reference?
I am just looking for a reference.
Given $\phi$, let $$ A=\{u\in W^{1,p};u-\phi\in W^{1,p}_0\} $$ be the admissible set. The solution $u$ is the unique minimizer of the Dirichlet energy $$ I(v)=\int_\Omega|\nabla v|^p $$ in the set $A$. (This is a standard result in calculus of variations.)
Now, let $M=\sup\phi$ and $m=\inf\phi$ and define $$ v(x) = \begin{cases} M, & u(x)>M\\ m, & u(x)<m\\ u(x) & \text{otherwise}. \end{cases} $$ Now $v\in W^{1,p}$ and also $v\in A$. One can check that $|\nabla v|\leq|\nabla u|$ pointwise, so $I(v)\leq I(u)$. Since $u$ is the unique minimizer of $I$ in $A$, we have $u=v$. This means that $u$ is bounded from above by $\sup\phi$ and from below by $\inf\phi$. In particular $u\in L^\infty$.
I suppose you could find the comparison principle for the $p$-Laplace equation in most PDE books that cover the equation.