This paper deals with the existence of positive radial solutions for the quasilinear system $\text{div}(|\nabla u_i|^{p-2}\nabla u_i)+\lambda f^i(u_1,\dots,u_n)=0$, $|x|\lt1$, $u_i(x)=0$, on $|x|=1$, $i=1,\dots,n$, $p\gt1$, $\lambda>0$, $x\in\mathbb{R}^N$. The $f^i$, $i=1,\dots,n$, are continuous and non-negative functions. Let $\bm{u}=(u_1,\dots,u_n)$, $\|\bm{u}\|=\sum_{i=1}^n|u_i|$,

$$ f_0^i=\lim_{\|\bm{u}\|\to0}\frac{f^i(\bm{u})}{\|\bm{u}\|^{p-1}}, $$

$i=1,\dots,n$, $\bm{f}=(f^1,\dots,f^n)$, $\bm{f}_0=\sum_{i=1}^nf_0^i$. We prove that the problem has a positive solution for sufficiently small $\lambda>0$ if $\bm{f}_0=\infty$. Our methods employ a fixed-point theorem in a cone.

We study conditions under which every delta-convex (d.c.) mapping is the difference of two continuous convex operators, and vice versa. In particular, we prove that each d.c. mapping $F:(a,b)\to Y$ is the difference of two continuous convex operators whenever $Y$ belongs to a large class of Banach lattices which includes all $L^{p}(\mu)$ spaces ($1\leq p\leq\infty$). The proof is based on a result about Jordan decomposition of vector-valued functions. New observations on Jordan decomposition of finitely additive vector-valued measures are also presented.

This paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.