Singularly perturbed elliptic problems in exterior domains. (Q5933785)

The paper deals with the following problem: NEWLINE\[NEWLINE-\varepsilon^2\Delta u+u=u^{p-1},\quad u>0\text{ in }\Omega,\quad u\in H^1_0 (\Omega) \tag{1}NEWLINE\]NEWLINE where \(\Omega\) is a domain in \(\mathbb{R}^N\) such that \(\mathbb{R}^N \setminus \Omega\) is a bounded open set, \(2<P<\frac{2N}{N-2}\) if \(N >2\) and \(2<p<+\infty\) if \(N=2\). The authors prove that (1) has no single peak solution if \(\mathbb{R}^N\setminus\Omega\) is convex. Moreover, they show that (1) always has a two-peak solution and to have a single peak solution depends on the domain topology or the domain geometry.