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On the average value for nonconstant eigenfunctions of the \(p\)-Laplacian assuming Neumann boundary data. (English) Zbl 1109.35371
Summary: We show that nonconstant eigenfunctions of the \(p\)-Laplacian do not necessarily have an average value of 0, as they must when \(p=2\). This fact has implications for deriving a sharp variational characterization of the second eigenvalue for a general class of nonlinear eigenvalue problems.

MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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