PROC
Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains
Takahiro Hashimoto
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.
keywords: singularity. nonexistence strip-like domains
PROC
Pohozaev-Ôtani type inequalities for weak solutions of quasilinear elliptic equations with homogeneous coefficients
Takahiro Hashimoto
In this paper, we are concerned with the following quasilinear el- liptic equations:
keywords: quasilinear elliptic equation nonexistence
PROC
Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients
Takahiro Hashimoto
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 $b$.

keywords: nonexistence elliptic equation quasilinear
DCDS
Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains
Takahiro Hashimoto Mitsuharu Ôtani
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 $q \le p$).
keywords: Quasilinear elliptic equations positive solutions.
PROC
Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations
Takahiro Hashimoto
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.
keywords: biharmonic nonexistence. existence
PROC
Nonexistence of global solutions of nonlinear Schrodinger equations in non star-shaped domains
Takahiro Hashimoto
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.
keywords: Blow-up nonlinear Sch¨odinger mixed problem.

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