On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem

Given a smooth bounded domain $ \Omega \subset \mathbb {R}^n $ and consider the problem \begin{document}$ \left\{\begin{array} {cccccc} - \Delta u = |u|^p - \sigma &\hbox{in } \Omega \\ \dfrac{\partial u}{\partial \nu} = 0 &\hbox{on}\ \partial \Omega \end{array}\right. $\end{document} where $ p $ is subcritical exponent ($ p > 1 $ if $ n = 2 $ and $ 1 < p < \frac{n+2}{n-2} $ if $ n \geq 3 $), $ \sigma > 0 $ is a large parameter and $ \nu $ denotes the outward normal of $ \partial\Omega $. Let $ \Gamma $ be an interior straighline intersecting orthogonally with $ \partial\Omega $. Assuming moreover that $ \Gamma $ satisfies a non-degeneracy condition, we construct a new class of solutions which consist of large number of spikes concentrating on $ \Gamma $, showing as in [5,6] that higher dimensional concentration can exist without resonance condition.