## Deformation of domain and the limit of the variational eigenvalues of semilinear elliptic operators.(English)Zbl 0865.35099

The author studies the following semilinear elliptic eigenvalue problem: $Lu+f(x,u)=0\quad\text{in }\Omega_r,\quad u=0\quad\text{on }\partial\Omega_r,$ where $$\Omega_r\subset\mathbb{R}^N$$ is a bounded domain with a smooth boundary with a positive parameter $$r$$ and $$L$$ is a second-order uniformly elliptic selfadjoint operator. The main problem is the domain dependency of the set of eigenvalues $$\{\mu_n(r,\alpha)\}^\infty_{n=1}$$ defined through the Lyusternik-Schnirelman theory. The author proves under a certain kind of smoothness of the deformation of $$\Omega_r$$ that $$\mu_n(r,\alpha)$$ varies continuously in $$r$$. The condition on $$\Omega_r$$ for $$\mu_n(r,\alpha)$$ to go to $$\infty$$ is also given.
Reviewer: S.Jimbo (Sapporo)

### MSC:

 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

domain dependency of eigenvalues
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