Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem
Weiwei Ao, Mengdie Fu, Chao Liu · Discrete and Continuous Dynamical Systems · 2022
Given a smooth bounded domain \begin{document}$ \Omega\subset{{\mathbb R}}^2 $\end{document} , we consider the following Ambrosetti-Prodi problem with Neumann boundary: \begin{document}$ \begin{equation*} \left\{\begin{array}{l} -\Delta u = \left\vert{u}\right\vert^p-\sigma \quad {\mbox {in}} \ \Omega,\\ {\partial u \over \partial u} = 0 \quad {\mbox {on}} \ \partial \Omega. \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ p>2 $\end{document} , \begin{document}$ \sigma>0 $\end{document} is a large parameter and \begin{document}$ u $\end{document} denotes the outward normal of \begin{document}$ \partial \Omega $\end{document} . We constructed a new class of solutions comprised of a large number of spikes concentrated on a segment of the boundary containing a local minimum point of the mean curvature function and having the same mean curvature at the endpoints. A similar boundary-concentrating phenomenon was obtained for the Lin-Ni-Takagi problem by Ao et al. [ 3 ].