Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. I: \(N=3\) (Q1883239)

In the present study the authors consider the nonlinear Neumann elliptic problem \[ \begin{cases} -\Delta u+\mu u=u^q,\quad & x\in\Omega\\ u> 0\quad & \text{in }\Omega\\ \frac{\partial u}{\partial u}=0\quad & \text{on }\partial\Omega,\end{cases}\tag{1} \] where \(1<q<+\infty\), \(\mu >0\) and \(\Omega\) is a smooth and bounded domain in \(\mathbb R^3\). The goal of the author is to study the problem for fixed \(\mu\), when exponent \(q\) is close to the critical one, that is \(q=5+\varepsilon\) and \(\varepsilon\) is a small nonzero number.