Multiplicity of single-bump solutions for semilinear elliptic equations in multi-bump domains (Q1888596)

The author studies the following semilinear elliptic equation \[ \begin{cases} -\Delta u+ u= |u|^{p-2}u\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\) is a domain in \(\mathbb{R}^N\) and \(p\) satisfies \(p> 2\) if \(N= 2\) and \(2< p<{2N\over N-2}\) for \(N\geq 3\). The main goal of the author is to use the shape of the domain \(\Omega\) prove that (1) has multiple positive solutions.