On the number of interior multipeak solutions for singularly perturbed Neumann problems (Q1284662)

The author estimates the number of solutions with exactly \(k\) interior local maximum points for the singularly perturbed problem \[ -\varepsilon^2\Delta u+ u= u^{p-1}\quad\text{in }\Omega, \] \[ u>0\quad\text{in }\Omega\tag{1} \] \[ {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega, \] where \(\Omega\subset \mathbb{R}^N\), \(1<p<(N+ 2)/(N- 2)\) if \(N\geq 3\) and \(p\in(1,\infty)\) if \(N= 2\). It is proved that (1) always has a solution \(u_\varepsilon\) with exactly \(k\) local maximum points lying in \(\Omega\).