Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains

Takahiro Hashimoto; Mitsuharu Ôtani

doi:10.3934/dcds.1997.3.565

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1997, Volume 3, Issue 4: 565-578. Doi: 10.3934/dcds.1997.3.565

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Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains

Takahiro Hashimoto^1, and
Mitsuharu Ôtani^2,

1.
Department of Mathematical Sciences, Faculty of Science, Ehime University, 2-5 Bunkyo-cho, Matsuyama-shi, Ehime, Japan 790-77
2.
Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555

Received: December 31, 1996

Revised: May 31, 1997

Published: September 1997

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Abstract

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.

Mathematics Subject Classification: 35J20, 35J70.

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