Nonexistence of weak  solutions of quasilinear ellipticequations with variable coefficients

Takahiro Hashimoto

doi:10.3934/proc.2009.2009.349

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2009, Volume 2009: 349-358. Doi: 10.3934/proc.2009.2009.349 open access

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Nonexistence of weak solutions of quasilinear elliptic equations with variable coefficients

Takahiro Hashimoto^1,

1.
Meteorological College, 7-4-81, Asahi0cho, Kashiwa, Chiba, 277-0852

Received: September 2008

Revised: April 2009

Published: August 2009

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Abstract

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 $b$.

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Mathematics Subject Classification: Primary: 35J25; Secondary: 35J20, 35J70.

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