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Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

Futoshi Takahashi

Department of Mathematics, Osaka City University and Osaka City University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan

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https://doi.org/10.1142/S0219199714500163Cited by:1 (Source: Crossref)

Abstract

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions

where Ω ⊂ ℝ^N, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

Keywords:

AMSC: 35J20, 35J25, 35J60

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