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### Open Access Journals

PROC

In this paper, we discuss the nonexistence of positive solutions
for nonlinear elliptic equations with singularity on the boundary in infinite
strip-like domains.
The main results are the nonexistence
in eigen-value and sub principal case and they are shown by giving an argument concerning the
simplicity of the first eigenvalue for generalized eigenvalue
problems combined with translation invariance of the domain. We also
show another nonexistence result via a modified
version of Pohozaev-type identity.

PROC

In this paper, we are concerned with the following quasilinear el-
liptic equations:

PROC

In this paper, we are concerned with the
following quasilinear elliptic equations:

-div${a(x)|\nabla u|^{p-2}\nabla u$ $=b(x)|u|^(q-2)u $ in $\Omega$

$u(x)$ $= 0$
on $\partial\Omega$

where $\Omega$ is a domain in $\mathbf R^N$ $(N \ge 1)$ with smooth
boundary.

When $a$ and $b$ are positive constants, there are many results on the
nonexistence of nontrivial solutions for the equation (E).
The main purpose of
this paper is to discuss the nonexistence results for
(E) with a class of weak solutions under some assumptions on $a$ and $b$.

DCDS

The nonexistence of positive solutions is discussed for $-\Delta_p u =
a(x) u^{q-1}$ in $\Omega$, $u|_{\partial\Omega}= 0$, for the case
where $a(x)$ is a bounded positive function and $\Omega$ is a strip-like domain such as $\Omega =
\Omega_d \times \mathcal{R}^{N-d}$ with $\Omega_d$ bounded in $\mathcal{R}^{d}$. The
existence of nontrivial solution of (E) is proved by Schindler for
$q \in (p, p*)$ where $p*$ is Sobolev's critical exponent.
Our method of proofs for nonexistence rely on the "Pohozaev-type
inequality" (for $q \ge p*$); and on a new argument concerning the
simplicity of the first eigenvalue for (generalized) eigenvalue
problems combined with translation invariance of
the domain (for $q \le p$).

PROC

Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations

In this paper, we are concerned with the existence and nonexistence of
nontrivial solutions for nonlinear elliptic equations involving a biharmonic operator.
Concerning the second order equations, a complementary result was obtained for the
problem of interior, exterior and whole space. The main purpose of this paper is
to discuss whether the complementary result mentioned above is still valid for the
nonlinear fourth order equations. We introduce "Kelvin type transformation" for
a biharmonic operator to convert an exterior problem to an interior problem. The
existence results in case of super-critical exterior problem are shown by introducing
a weighted version of Sobolev-Poincaré type inequality, and the nonexistence results
are shown by giving a Pohozaev-type identity for fourth order equations.

PROC

In this paper, we discuss the nonexistence of global solutions of mixed problems of the nonlinear Schödinger equations with power nonlinearity. When the domain is whole space, there are many results concerning the nonexistence of global solutions ( or existence of blow-up solutions ) for the equation. For the case of a general domain, there are few studies of blowing-up
conditions. The main purpose of this paper is to discuss the nonexistence of global solutions in a deformed tube-shaped domain which is not star-shaped.

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