# American Institute of Mathematical Sciences

## Journals

CPAA
Communications on Pure & Applied Analysis 2010, 9(1): 109-140 doi: 10.3934/cpaa.2010.9.109
We establish Hardy inequalities of the form

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

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.

keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2019, 39(5): 2731-2742 doi: 10.3934/dcds.2019114
In this paper we consider positive supersolutions of the nonlinear elliptic equation
 $- \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega,$
where
 $0\le p<1$
,
 $\Omega$
is an arbitrary domain (bounded or unbounded) in
 ${\mathbb{R}}^N$
(
 $N\ge 2$
),
 $f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+}$
 $(0 < a_{f} \leq +\infty)$
is a non-decreasing continuous function and
 $\rho: \Omega \rightarrow \mathbb{R}$
is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions
 $u$
at each point
 $x\in\Omega$
where
 $\nabla u\not\equiv0$
in a neighborhood of
 $x$
. As applications, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains
 $\Omega$
with the property that
 $\sup_{x\in\Omega}dist (x, \partial\Omega) = \infty$
. In particular when
 $\rho(x) = |x|^\beta$
(
 $\beta\in {\mathbb{R}}$
) and
 $f(u) = u^q$
with
 $q+p>1$
then every positive supersolution in an exterior domain is eventually constant if
 $(N-2)q+p(N-1)< N+\beta.$
keywords:
DCDS
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1033-1050 doi: 10.3934/dcds.2010.28.1033
We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $\R^N$, with the Navier boundary condition $u=\Delta u =0$ on δΩ. We establish energy estimates which show that for any non-decreasing convex and superlinear nonlinearity $f$ with $f(0)=1$, the extremal solution u * 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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