We study convergence properties of ${\left\{v\left(\nabla u_k\right)\right\}}_{k\in \mathbb{N}}$ if $v\in C\left({ℝ}^{m\times n}\right)$, $\left|v\left(s\right)\right|\le C(1+|s{|}^p)$, $1<p<+\infty$, has a finite quasiconvex envelope, $u_k\to u$ weakly in $W^{1,p}(\Omega ;{ℝ}^m)$ and for some $g\in C\left(\Omega \right)$ it holds that ${\int }_{\Omega }g\left(x\right)v\left(\nabla u_k\left(x\right)\right)\mathrm{d}x\to {\int }_{\Omega }g\left(x\right)Qv\left(\nabla u\left(x\right)\right)\mathrm{d}x$ as $k\to \infty$. In particular, we give necessary and sufficient conditions for $L^1$-weak convergence of ${\{det\nabla u_k\}}_{k\in \mathbb{N}}$ to $det\nabla u$ if $m=n=p$.

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator $W_k:=\Delta +\frac{k}{x_n}\frac{\partial }{\partial x_n}$ on ${ℝ}^n$ is proved. In this note there is shown that in the cases $k\ne 0$, $k\ne 2$ no other transforms of this kind exist and for case $k=2$, all such transforms are described.

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems $$\left\}\begin{array}{cccc}-{\Delta }_{p}u=f(x,u,v),\hfill & \text{in}\Omega ,-{\Delta }_{p}v=g(x,u,v),\hfill & \text{in}\Omega ,u=v=0,& \text{on}\partial \Omega ,\end{array}\right.$$ where $-{\Delta }_p$ is the $p$-Laplace operator, $p>1$ and $\Omega$ is a $C^{1,\alpha }$-domain in ${ℝ}^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)={\lambda }_1{v}^{p-1}$, $g(x,u,v)={\lambda }_2{u}^{p-1}$ and obtain a non-existence result of positive solutions for the general systems.