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Superlinear problems on domains with holes of asymptotic shape and exterior problems. (English) Zbl 0933.35068

The paper deals with the problem \[ \begin{cases} -\nabla u=u^p\quad & \text{in }D \setminus U_n\\ u=0\quad & \text{on }\partial(D\setminus U_n)\\ u>0 \quad & \text{in }D\setminus U_n \end{cases} \tag{1} \] where \(1<p<(m+2)(m-2)^{-1} \), \(D\) is a bounded domain in \(\mathbb{R}^m\) with smooth boundary and \(U_n\) are small open holes (not necessarily connected). In his previous paper the author showed that if \(U_n\) consists of a finite number of star shaped holes (with number independent on \(n)\) not close to the boundary or to each other and if the diameter of each star shaped hole tends to zero with \(n\), then there is a uniform in \(n\) bound for the positive solutions of (1). The interest in this is that it then follows easily from domain variation theory that for large \(n\) a positive solution of (1) is close to a positive solution of \[ \begin{cases} -\nabla u=u^p \quad & \text{in }D\\ u=0\quad & \text{on }D\\ u>0\quad & \text{in }D\end{cases}.\tag{2} \] In this paper the case where the small holes have some limiting shape is considered.
Reviewer: V.Mustonen (Oulu)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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