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$ \int_\Omega | \nabla u|_A^2 dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2}u^2dx, \qquad u \in H_0^1(\Omega) \qquad\qquad (1)$

where $ E$ is a positive function defined in $ \Omega$, -div$(A \nabla E)$ is a nonnegative nonzero finite measure in $ \Omega$ which we denote by $ \mu$ and where $ A(x)$ is a $ n \times n$ symmetric, uniformly positive definite matrix defined in $ \Omega$ with $ | \xi |_A^2:= A(x) \xi \cdot \xi$ for $ \xi \in \mathbb{R}^n$. We show that (1) is optimal if $ E=0$ on $ \partial \Omega$ or $ E=\infty$ on the support of $ \mu$ and is not attained in either case. When $ E=0$ on $\partial \Omega$ we show

$ \int_\Omega | \nabla u|_A^2dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2}u^2dx + \frac{1}{2} \int_\Omega \frac{u^2}{E} d \mu, \qquad u \in H_0^1(\Omega)\qquad (2) $

is optimal and not attained.
Optimal weighted versions of these inequalities are also established. Optimal analogous versions of (1) and (2) are established for $p$≠ 2 which, in the case that $ \mu$ is a Dirac mass, answers a best constant question posed by Adimurthi and Sekar (see [1]).

We examine improved versions of the above inequalities of the form

$\int_\Omega | \nabla u|_A^2dx \ge \frac{1}{4} \int_\Omega \frac{| \nabla E|_A^2}{E^2} u^2dx + \int_\Omega V(x) u^2dx, \qquad u \in H_0^1(\Omega).\qquad (3)$

Necessary and sufficient conditions on $V$ are obtained (in terms of the solvability of a linear pde)
for (3) to hold. Analogous results involving improvements are obtained for the weighted versions.

In addition we obtain various results concerning the above inequalities valid for functions $ u$ which are nonzero on the boundary of $ \Omega$. We also examine the nonquadradic case ,ie. $p$ ≠2 of the above inequalities.

$ - \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega, $ |

$ 0\le p<1 $ |

$ \Omega $ |

$ {\mathbb{R}}^N $ |

$ N\ge 2 $ |

$ f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+} $ |

$ (0 < a_{f} \leq +\infty) $ |

$ \rho: \Omega \rightarrow \mathbb{R} $ |

$ u $ |

$ x\in\Omega $ |

$ \nabla u\not\equiv0 $ |

$ x $ |

$ \Omega $ |

$ \sup_{x\in\Omega}dist (x, \partial\Omega) = \infty $ |

$ \rho(x) = |x|^\beta $ |

$ \beta\in {\mathbb{R}} $ |

$ f(u) = u^q $ |

$ q+p>1 $ |

$ (N-2)q+p(N-1)< N+\beta. $ |

^{*}is smooth provided $N\leq 5$. If in addition $\lim$i$nf_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}>0$, then u

^{*}is regular for $N\leq 7$, while if $\gamma$:$= \lim$s$up_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}<+\infty$, then the same holds for $N < \frac{8}{\gamma}$. It follows that u

^{*}is smooth if $f(t) = e^t$ and $ N \le 8$, or if $f(t) = (1+t)^p$ and $N< \frac{8p}{p-1}$. We also show that if $ f(t) = (1-t)^{-p}$, $p>1$ and $p\ne 3$, then u

^{*}is smooth for $N \leq \frac{8p}{p+1}$. While these results are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established on radial domains, e.g., u

^{*}is smooth for $ N \le 12$ when $ f(t) = e^t$ [11], and for $ N \le 8$ when $ f(t) = (1-t)^{-2}$ [9] (see also [22]).

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