### Abstract

We consider the following elliptic problem with a nonlinear boundary condition: - Δ u + b u = | u |^{p - 1} u in Ω, - frac(∂ u, ∂ n) = | u |^{q - 1} u - g (u) on ∂ Ω, where 1 < q < p and p < (N + 2) / (N - 2), if N ≥ 3. The existence of solutions to this problem is discussed under suitable conditions on g (u). Our proof relies on the variational argument. However, since L^{q + 1} (∂ Ω) ⊂ H^{1} (Ω) does not hold for large q, we cannot apply the variational method in a direct way. To overcome this difficulty, some approximation problems are introduced and uniform a priori estimates for solutions of approximate equations are established.

Original language | English |
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Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 71 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2009 Dec 15 |

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### Keywords

- Nonlinear boundary condition
- Variational problem

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Semilinear elliptic equations with nonlinear boundary conditions.** / Harada, Junichi; Otani, Mitsuharu.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Semilinear elliptic equations with nonlinear boundary conditions

AU - Harada, Junichi

AU - Otani, Mitsuharu

PY - 2009/12/15

Y1 - 2009/12/15

N2 - We consider the following elliptic problem with a nonlinear boundary condition: - Δ u + b u = | u |p - 1 u in Ω, - frac(∂ u, ∂ n) = | u |q - 1 u - g (u) on ∂ Ω, where 1 < q < p and p < (N + 2) / (N - 2), if N ≥ 3. The existence of solutions to this problem is discussed under suitable conditions on g (u). Our proof relies on the variational argument. However, since Lq + 1 (∂ Ω) ⊂ H1 (Ω) does not hold for large q, we cannot apply the variational method in a direct way. To overcome this difficulty, some approximation problems are introduced and uniform a priori estimates for solutions of approximate equations are established.

AB - We consider the following elliptic problem with a nonlinear boundary condition: - Δ u + b u = | u |p - 1 u in Ω, - frac(∂ u, ∂ n) = | u |q - 1 u - g (u) on ∂ Ω, where 1 < q < p and p < (N + 2) / (N - 2), if N ≥ 3. The existence of solutions to this problem is discussed under suitable conditions on g (u). Our proof relies on the variational argument. However, since Lq + 1 (∂ Ω) ⊂ H1 (Ω) does not hold for large q, we cannot apply the variational method in a direct way. To overcome this difficulty, some approximation problems are introduced and uniform a priori estimates for solutions of approximate equations are established.

KW - Nonlinear boundary condition

KW - Variational problem

UR - http://www.scopus.com/inward/record.url?scp=72149131748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72149131748&partnerID=8YFLogxK

U2 - 10.1016/j.na.2009.09.013

DO - 10.1016/j.na.2009.09.013

M3 - Article

VL - 71

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 12

ER -