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### Open Access Journals

DCDS-B

We show that for a very general and natural class of curvature functions (for example the curvature quotients
$(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface
in de Sitter space satisfying
$f(\kappa)=\sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity
has at least one smooth solution (if $l=1$ or $l=2$ there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space
and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l=\sigma,\,1 \leq l < n$ in both de Sitter and Hyperbolic space.

DCDS

It is well known through the work of Majumdar, Papapetrou, Hartle, and Hawking that the coupled
Einstein and Maxwell equations admit a static multiple blackhole solution representing a balanced equilibrium state of finitely many point charges. This is a result of the exact cancellation of gravitational attraction and electric repulsion under an explicit condition on the mass and charge ratio.
The resulting system of particles, known as an

*extremely charged*dust, gives rise to examples of spacetimes with naked singularities. In this paper, we consider the continuous limit of the Majumdar-Papapetrou-Hartle-Hawking solution modeling a space occupied by an extended distribution of extremely charged dust. We show that for a given smooth distribution of matter of finite ADM mass there is a continuous family of smooth solutions realizing asymptotically flat space metrics.
DCDS

We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary
condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive,
nondecreasing, and convex nonlinearity which is superlinear at infinity.
Under these assumptions, the boundedness of all semistable solutions
is expected up to dimension $n\leq 9$, but only established for $n\leq 4$.

In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound will follow from a $L^p$-estimate for $f'(u)$ for every $p<3$ (and for all $n\geq 2$). Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.

In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound will follow from a $L^p$-estimate for $f'(u)$ for every $p<3$ (and for all $n\geq 2$). Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.

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