Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem

Abstract

<p style='text-indent:20px;'>Given a smooth bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset{{\mathbb R}}^2 $\end{document}</tex-math></inline-formula>, we consider the following Ambrosetti-Prodi problem with Neumann boundary:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{l} -\Delta u = \left\vert{u}\right\vert^p-\sigma \quad {\mbox {in}} \ \Omega,\\ {\partial u \over \partial \nu} = 0 \quad {\mbox {on}} \ \partial \Omega. \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ p&gt;2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \sigma&gt;0 $\end{document}</tex-math></inline-formula> is a large parameter and <inline-formula><tex-math id="M4">\begin{document}$ \nu $\end{document}</tex-math></inline-formula> denotes the outward normal of <inline-formula><tex-math id="M5">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula>. 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. [<xref ref-type="bibr" rid="b3">3</xref>].</p>