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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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