## 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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