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Estimates of the principal eigenvalue of the \(p\)-Laplacian and the \(p\)-biharmonic operator. (English) Zbl 1349.35133

Summary: We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet \(p\)-Laplacian and the Navier \(p\)-biharmonic operator on a ball of radius \(R\) in \(\mathbb{R}^N\) and its asymptotics for \(p\) approaching 1 and \(\infty\).
Let \(p\) tend to \(\infty\). There is a critical radius \(R_C\) of the ball such that the principal eigenvalue goes to \(\infty\) for \(0<R\leq R_C\) and to \(0\) for \(R>R_C\). The critical radius is \(R_C=1\) for any \(N\in\mathbb{N}\) for the \(p\)-Laplacian and \(R_C=\sqrt{2N}\) in the case of the \(p\)-biharmonic operator.
When \(p\) approaches 1, the principal eigenvalue of the Dirichlet \(p\)-Laplacian is \(NR^{-1}\times (1-(p-1)\log R(p-1))+o(p-1)\) while the asymptotics for the principal eigenvalue of the Navier \(p\)-biharmonic operator reads \(2N/R^2+O(-(p-1)\log (p-1))\).

MSC:

35J66 Nonlinear boundary value problems for nonlinear elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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