Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w*}(X*,Y)$, as well as the complementation of the space $K_{w*}(X*,Y)$ of w*-w compact operators in the space $L_{w*}(X*,Y)$ of w*-w operators from X* to Y.

We will consider the following problem$$-\Delta u-\lambda \frac{u}{{\left|x\right|}^2}={\left|\nabla u\right|}^p+cf,u\>0\text{in}\Omega ,$$where $\Omega \subset {ℝ}^N$ is a domain such that $0\in \Omega$, $N\ge 3$, $c\>0$ and $\lambda \>0$. The main objective of this note is to study the precise threshold $p_{+}=p_{+}\left(\lambda \right)$ for which there is novery weak supersolutionif $p\ge p_{+}\left(\lambda \right)$. The optimality of $p_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p\<p_{+}\left(\lambda \right)$, for $c\>0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

We establish the Euler-Lagrange inclusion of a nonsmooth integral functional defined on Orlicz-Sobolev spaces. This result is achieved through variational techniques in nonsmooth analysis and an integral representation formula for the Clarke generalized gradient of locally Lipschitz integral functionals defined on Orlicz spaces.

Let Φ be an N-function, then the Jung constants of the Orlicz function spaces LΦ[0,1] generated by Φ equipped with the Luxemburg and Orlicz norms have the exact value:(i) If FΦ(t) = tφ(t)/Φ(t) is decreasing and 1 < CΦ < 2, then JC(L(Φ)[0,1]) = JC(LΦ[0,1]) = 21/CΦ-1;(ii) If FΦ(t) is increasing and CΦ > 2, then JC(L(Φ)[0,1]) = JC(LΦ[0,1])=2-1/CΦ,where CΦ= limt→+∞ tφ(t)/Φ(t).

We give conditions under which the functor projective limit is exact on the category of quotients of Fréchet spaces of L. Waelbroeck [18].

Let $D$ be a domain in $C^2$. Given $w\in C$, set $D_w=\left\{z\in C\mid \left(z,w\right)\in D\right\}$. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E\left(D,f\right)$ of all $w$ such that $f(\cdot ,w)$ is not square-integrable in $D_w$ has measure zero. We call this set the exceptional set for $f$. In this Note we prove that whenever $0<r<1$ there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $C^2$ such that $E\left(B,f\right)$ is the circle $C\left(0,r\right)=\left\{z\in C\mid \left|z\right|=r\right\}$.

The aim of this paper is to start a systematic investigation of the existence of angular limits and angular derivatives of holomorphic maps of infinite dimensional Siegel domains in $J^{*}$-algebras. Since $J^{*}$-algebras are natural generalizations of $C^{*}$-algebras, $B^{*}$-algebras, $J{C}^{*}$-algebras, ternary algebras and complex Hilbert spaces, various significant results follow. Examples are given.