Laplace's equation with concave and convex boundary nonlinearities on an exterior region

2017/08/21 by Mao, Jinxiu, Zhao, Zengqin
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1708.06066

Abstract

This paper studies Laplace's equation -Δ u=0 in an exterior region U\varsubsetneq\mathbb R^N, when N≥3, subject to the nonlinear boundary condition (∂ u)/(∂ν)=λ\vertu\vert^q-2u+μ\vertu\vert^p-2u on ∂ U with 10 and μ∈\mathbb R arbitrary, then there exists a sequence \u_k\ of solutions with negative energy converging to 0 as k→∞; on the other hand, when λ∈\mathbb R and μ>0 arbitrary, then there exists a sequence \u_k\ of solutions with positive and unbounded energy. Also, associated with the p-Laplacian equation -Δ_p u=0, the exterior p-harmonic Steklov eigenvalue problems are described.

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