Superlinear elliptic Dirichlet problems in almost spherically symmetric exterior domains (Q1114855)

The authors study the existence of positive solutions of the problem \[ \Delta u-u+f(u)=0 \text{ in } \Omega;\quad u=0 \text{ on } \partial \Omega, \tag{1} \] where \(\Omega\) is an exterior domain in \(R^3\) (the complement of a nearly spherical domain) and f is a ``superlinear'' function vanishing at zero to order higher than the first. This problem gives rise to the consideration of two additional questions: the structure of positive, radially symmetric solutions of (1) which vanish at infinity and the study of the properties of the spectrum of the eigenvalue problem \[ \Delta v-v-\lambda f'(u)v=0 \text{ in } \Omega_ R, \quad v\in \overset\circ W_{1,2}(\Omega_ R); \quad v=0 \text{ on } \partial \Omega_ R, \] where \(\Omega_ R=\{x\in R^ 3: | x| >R\}\) and \(u\) is a positive radially symmetric solution of (1) with \(\Omega =\Omega_ R.\) For the proof of their results, the authors use a perturbation argument and the approach to the eigenvalue problem is realized using the expansion of v into a series of spherical harmonics.