## 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

### Keywords:

domain variation theory; positive solution; limiting shape
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