### 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 |

### Keywords

- Nonlinear boundary condition
- Variational problem

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

Harada, J., & Otani, M. (2009). Semilinear elliptic equations with nonlinear boundary conditions.

*Nonlinear Analysis, Theory, Methods and Applications*,*71*(12). https://doi.org/10.1016/j.na.2009.09.013