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Asymptotically radial solutions in expanding annular domains. (English) Zbl 1276.35083

Summary: In this paper, we consider the problem \[ \begin{cases} -\Delta u=u^{p}\quad&\text{in }\Omega_R,\\ u=0 \quad&\text{on }\partial\Omega_R,\end{cases}\tag{0.1} \] where \(p>1\) and \(\Omega_R\) is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as \(R \longrightarrow \infty\). The main goal of the paper is to prove, for large \(R\), the existence of a positive solution to (0.1) which is close to the positive radial solution in the corresponding diffeomorphic annulus. The proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions.

MSC:

35J61 Semilinear elliptic equations
35B09 Positive solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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