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Existence and starshapedness for the Lane-Emden equation. (English) Zbl 1089.35022

The paper deals with the equation \(\Delta u+u^{p}=0\) with supercritical \( p\geq \left( N+2\right) /\left( N-2\right) ,\) \(N\geq 3\) in a starshaped ring \(\Omega ,\) i.e., \(\Omega =\Omega _{0}\setminus \overline{\Omega _{1}}\) with bounded domains \(\Omega _{1}\subset \subset \Omega _{0}\subset \mathbb{R}^{N} \) which are starshaped with respect to the same point. It is proved that if \( \Omega \) is regular and \(\Omega _{1}\) is convex, then there exists at least one positive solution with starshaped level sets and prescribed constant values on \(\partial \Omega _{0}\) and \(\partial \Omega _{1}.\) The results are achieved by means of the Kelvin transformation and the method of moving spheres.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J60 Nonlinear elliptic equations
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