The Hardy inequality ${\int }_{\Omega }{\left|u\left(x\right)\right|}^{p}d{\left(x\right)}^{-p}\ddot{x}\le c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^p\ddot{x}$ with $d\left(x\right)=dist(x,\partial \Omega )$ holds for $u\in C_0^{\infty }\left(\Omega \right)$ if $\Omega \subset {ℝ}^n$ is an open set with a sufficiently smooth boundary and if $1<p<\infty$. P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p=1$.

In this paper existence and multiplicity of solutions of the elliptic problem $ℒu+{\lambda }_{1}u+\mu u^{+}v{u}^{-}+g(x,u)=f$ in $\Omega$ $Bu=0$ on $\partial \Omega$, are discussed provided the parameters $\mu$ and $v$ are close to the first eigenvalue ${}_1$. The sufficient conditions presented here are more general than those in given by S. Fučík in his aerlier paper.

We show a locally uniform bound for global nonnegative solutions of the system $u_t=\Delta u+uv-bu$, $v_t=\Delta v+au$ in $(0,+\infty )\times \Omega$, $u=v=0$ on $(0,+\infty )\times \partial \Omega$, where $a>0$, $b\ge 0$ and $\Omega$ is a bounded domain in ${ℝ}^n$, $n\le 2$. In particular, the trajectories starting on the boundary of the domain of attraction of the zero solution are global and bounded.