### Abstract

The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = R^{N} \Ω_{0}, with Ω_{0} bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a "Pohozaev-type inequality" valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.

Original language | English |
---|---|

Pages (from-to) | 267-290 |

Number of pages | 24 |

Journal | Houston Journal of Mathematics |

Volume | 23 |

Issue number | 2 |

Publication status | Published - 1997 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Houston Journal of Mathematics*,

*23*(2), 267-290.

**Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains.** / Hashimoto, Takahiro; Otani, Mitsuharu.

Research output: Contribution to journal › Article

*Houston Journal of Mathematics*, vol. 23, no. 2, pp. 267-290.

}

TY - JOUR

T1 - Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains

AU - Hashimoto, Takahiro

AU - Otani, Mitsuharu

PY - 1997

Y1 - 1997

N2 - The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = RN \Ω0, with Ω0 bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a "Pohozaev-type inequality" valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.

AB - The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = RN \Ω0, with Ω0 bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a "Pohozaev-type inequality" valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.

UR - http://www.scopus.com/inward/record.url?scp=0031475170&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031475170&partnerID=8YFLogxK

M3 - Article

VL - 23

SP - 267

EP - 290

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

SN - 0362-1588

IS - 2

ER -