## Journals

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### Open Access Journals

CPAA

In the paper, we apply the moving plane method to prove that if the right hand sides of equation and Neumann boundary condition are both independent of one variable, the domain and the solution to the Hessian over-determined problem are mirror symmetric. Our result generalizes the previous results on radial symmetry. In the end, we get the mirror symmetry of over-determined problems for more general equations, which include Weingarten curvature equation.

DCDS

This paper is mainly concerned with Euler-Poisson equations modeling
Newtonian stars. We establish the existence of rotating star
solutions for general compressible fluids with prescribed angular
velocity law, which is the main point distinguished with the case
with prescribed angular momentum per unit mass. The compactness of
any minimizing sequence is established, which is important from the
stability point of view.

CPAA

In this paper, we consider the obstacle problem for Monge-Ampère
type equations which include prescribed Gauss curvature equation as
a special case. We establish $C^{1,1}$ regularity of the greatest
viscosity solution in non-convex domains.

DCDS

In the paper, we give the positive answer of an open problem of
Li-Nirenberg under the weaker conditions, and we prove a new
variation of the boundary point lemma for second order fully
nonlinear ODEs by a new method. A simpler proof of Li-Nirenberg
Theorem is also presented.

DCDS

In this paper, we apply the sharp Adams-type inequalities for the Sobolev
space $W^{m,\frac{n}{m}}\left(
\mathbb{R}
^{n}\right) $ for any positive real number $m$ less than $n$, established by Ruf and Sani [46] and Lam and Lu [30,31], to study
polyharmonic equations in $\mathbb{R}^{2m}$. We will consider the polyharmonic
equations in $\mathbb{R}^{2m}$ of the form
\[
\left( I-\Delta\right) ^{m}u=f(x,u)\text{ in }%
\mathbb{R}
^{2m}.
\]
We study the existence of the nontrivial solutions when the nonlinear terms
have the critical exponential growth in the sense of Adams' inequalities on the entire Euclidean space. Our
approach is variational methods such as the Mountain Pass Theorem ([5]) without
Palais-Smale condition combining with a version of a result due to Lions ([39,40]) for
the critical growth case. Moreover, using the regularity lifting by
contracting operators and regularity lifting by combinations of contracting
and shrinking operators developed in [14] and [11], we will prove
that our solutions are uniformly bounded and Lipschitz continuous. Finally,
using the moving plane method of Gidas, Ni and Nirenberg [22,23] in integral form developed by Chen, Li and Ou [12] together with the Hardy-Littlewood-Sobolev type inequality instead of the maximum
principle, we prove our positive solutions are radially symmetric and
monotone decreasing about some point. This appears to be the first work
concerning existence of nontrivial nonnegative solutions of the Bessel type
polyharmonic equation with exponential growth of the nonlinearity in the whole
Euclidean space.

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