2 problems

Let $p>1$, $n\geq 2$, and $r>\max\{p-1,1\}$. Let $u\in W^{1,r}_0(\mathbb{B}^n)\cap W^{1,p}_0(\mathbb{B}^n)$ and $f\in L^{\frac{r}{p-1}}\cap L^{p'}(\mathbb{B}^n,\mathbb{R}^n)$, wher…

Let $d,n\in\mathbb{N}$ and $1<p<\infty$. Let $h\colon\Omega\to\mathbb{R}^n$ be $p$-harmonic on an open set $\Omega\subset\mathbb{R}^d$, and let $V$ denote the standard nonlinear tr…