### Abstract

The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as -Δu = u ^{β} in Ω, ((N + 2)/(N - 2) < β), u = 0 on ∂B, with Ω = ℝ^{N}\Ω̄ where Ω_{0} is the open ball. To recover the compactness of the embedding L ^{β+1}(Ω) ⊂ H_{0}
^{1}(Ω, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.

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

Pages (from-to) | 323-333 |

Number of pages | 11 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 19 |

Issue number | 2 |

Publication status | Published - 2007 Oct |

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### Keywords

- Exterior domain
- P-Laplacian
- Supercritical exponent

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics
- Analysis

### Cite this

**Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains.** / Hashimoto, Satoshi; Otani, Mitsuharu.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems*, vol. 19, no. 2, pp. 323-333.

}

TY - JOUR

T1 - Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains

AU - Hashimoto, Satoshi

AU - Otani, Mitsuharu

PY - 2007/10

Y1 - 2007/10

N2 - The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as -Δu = u β in Ω, ((N + 2)/(N - 2) < β), u = 0 on ∂B, with Ω = ℝN\Ω̄ where Ω0 is the open ball. To recover the compactness of the embedding L β+1(Ω) ⊂ H0 1(Ω, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.

AB - The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as -Δu = u β in Ω, ((N + 2)/(N - 2) < β), u = 0 on ∂B, with Ω = ℝN\Ω̄ where Ω0 is the open ball. To recover the compactness of the embedding L β+1(Ω) ⊂ H0 1(Ω, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.

KW - Exterior domain

KW - P-Laplacian

KW - Supercritical exponent

UR - http://www.scopus.com/inward/record.url?scp=36248976091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36248976091&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:36248976091

VL - 19

SP - 323

EP - 333

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

SN - 1078-0947

IS - 2

ER -