Blowing up solutions for Neumann problem with critical nonlinearity on the boundary (Q304512)

In this paper, the authors deals with the following Neumann problem: \[ \begin{cases} \Delta u = 0,\qquad u > 0,\qquad&\text{in }\mathbb{R}_{+}^{n},\\ \displaystyle - \frac{\partial u}{\partial x_n} = K(x) u^{p - \varepsilon}\qquad&\text{on }\mathbb{R}_{+}^{n-1}, \end{cases}\tag{1} \] where \(n \geq 3\), \(\varepsilon\) is a small positive parameter, and \(K\) is a positive and bounded function in \(\mathbb{R}_{+}^{n-1}\). Using the variational method, the authors prove the existence of a peak solution that concentrate around one critical point of the function \(K\) satisfying certain conditions.