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.

Two new assertions characterizing analytically disks in the Euclidean plane ${ℝ}^2$ are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

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.