Let be the best Sobolev embedding constant of , where is a smooth bounded domain in . We prove that as the sequence converges to a constant independent of the shape and the volume of , namely 1. Moreover, for any sequence of eigenfunctions (associated with ), normalized by , there is a subsequence converging to a limit function which satisfies, in the viscosity sense, an -Laplacian equation with a boundary condition.
Submitted August 4, 2006. Published September 18, 2006.
Math Subject Classifications: 35J50, 35J55, 35J60, 35J65, 35P30.
Key Words: Nonlinear elliptic equations; eigenvalue problems; p-Laplacian; nonlinear boundary condition; Steklov problem; viscosity solutions.
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| An Lê |
Department of Mathematics and Statistics
Utah State University
Logan, Utah 84322, USA
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