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On the strict monotonicity of the first eigenvalue of the \(p\)-Laplacian on annuli. (English) Zbl 1405.35064
The paper deals with the nonlinear eigenvalue problem \(-\Delta_pu=\lambda|u|^{p-2}u\) for the \(p\)-Laplace operator in \(B_{R_1}(0)\setminus B_{R_0}(se_1)\subset\mathbb{R}^N\) with Dirichlet boundary conditions. Here \(1 < p < \infty\) and \(R_0+s < R_1\). The authors are interested in the first eigenvalue \(\lambda_1(p,s)\) as a function of \(s\). The first main result states that \(\partial_s\lambda_1<0\) for \(0 < s < R_1-R_0\). Next the authors deal with the limits \(\Lambda_\infty(s):=\lim_{p\to\infty}\lambda_1^{1/p}\lambda_1(p,s)\) and \(\Lambda_1(s):=\lim_{p\to1}\lambda_1(p,s)\) which are known to exist. The second main theorem states that \(\Lambda_\infty(s)\) is strictly decreasing in \(s\), and \(\Lambda_1(s)\) is decreasing and not constant. A corollary is that any eigenfunction associated to a point on the first nontrivial curve of the Fuçik spectrum of \(-\Delta_p\) in a bounded radial domain with Dirichlet boundary conditions is nonradial.

MSC:
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
49R05 Variational methods for eigenvalues of operators
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