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### Open Access Journals

DCDS

The blowup behaviors
of solutions to a scalar-field equation with the Robin condition
are discussed. For some range of the parameter, there exist at least
two positive solutions to the equation. Here, the blowup
rate of the large solution and the scaling properties are discussed.

CPAA

In this article, we consider the following semilinear elliptic equation on the hyperbolic space
\begin{eqnarray}
\Delta_{H^n} u-\lambda u+|u|^{p-1}u=0\quad on\quad H^n\setminus \{Q\}
\end{eqnarray}
where $\Delta_{H^n}$ is the Laplace-Beltrami operator on the hyperbolic space
\begin{eqnarray}
H^n=\{(x_1,\cdots, x_n,x_{n+1})|x_1^2+\cdots+x_n^2-x_{n+1}^2=-1\},
\end{eqnarray}
$n>10,\ p>1, \lambda>0, $ and $Q=(0,\cdots,0,1)$.
We provide the existence and uniqueness of a singular positive ``radial'' solution of the above equation for big $p$ (greater than the Joseph-Lundgren exponent, which appears if $n > 10$) as well as its asymptotic behavior.

DCDS-S

We consider the Cauchy problem of the two dimensional heat equation with a radially symmetric,
negative potential $-V$ which behaves like $V(r)=O(r^{-\kappa})$ as $r\to\infty$,
for some $\kappa > 2$. We study the rate and the direction for hot spots to tend to the spatial infinity.
Furthermore we give a sufficient condition for hot spots to consist of only one point
for any sufficiently large $t>0$.

CPAA

We propose a method to investigate the structure of positive radial
solutions to semilinear elliptic problems with various boundary conditions.
It is already shown that the boundary value problems can be reduced to a
canonical form by a suitable change of variables. We show structure theorems
to canonical forms to equations with power nonlinearities and various boundary
conditions. By using these theorems, it is possible to study the properties of
radial solutions of semilinear elliptic equations in a systematic way, and make
clear unknown structure of various equations.

CPAA

We consider elliptic boundary value problems on large spherical
caps with parameter dependent power nonlinearities. In this paper we show
that imperfect bifurcation occurs as in the work [13]. When the domain is the
whole sphere, there is a constant solution. In the case where the domain is a
spherical cap, however, the constant solution disappears due to the boundary
condition. For large spherical caps we construct solutions which are close to the
constant solution in the whole n-dimensional sphere, using the eigenvalues of
the linearized problem in the whole sphere and fixed point arguments based on a
Lyapunov-Schmidt type reduction. Numerically there is a surprising similarity
between the diagrams of this problem and the ones obtained in [18], also [5],
for a Brezis-Nirenberg type problem on spherical caps.

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