Semilinear elliptic equations with nonlinear boundary conditions

Junichi Harada, Mitsuharu Otani

    Research output: Contribution to journalArticle

    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 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.

    Original languageEnglish
    JournalNonlinear Analysis, Theory, Methods and Applications
    Volume71
    Issue number12
    DOIs
    Publication statusPublished - 2009 Dec 15

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    Keywords

    • Nonlinear boundary condition
    • Variational problem

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

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