## On the eigenvalues of the $$p$$-Laplacian with varying $$p$$.(English)Zbl 0882.35087

Summary: We study the nonlinear eigenvalue problem $-\text{div}(|\nabla u|^{p-2}\nabla u)= \lambda|u|^{p-2}u\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega,$ where $$p\in(1,\infty)$$, $$\Omega$$ is a bounded smooth domain in $$\mathbb{R}^N$$. We prove that the first and the second variational eigenvalues of (1) are continuous functions of $$p$$. Moreover, we obtain the asymptotic behavior of the first eigenvalue as $$p\to1$$ and $$p\to\infty$$.

### MSC:

 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

asymptotic behavior of the first eigenvalue
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### References:

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