Solutions with clustering concentration layers to the Ambrosetti-Prodi type problem
Qiang Ren
TL;DR
This work advances the nonlinear elliptic theory for Ambrosetti–Prodi type problems in the plane by constructing solutions with multiple concentration layers distributed along a closed curve $\Gamma$. Using an anisotropic, matrix-weighted operator $-\mathrm{div}(A(x)\nabla u)$ and a modified geometric framework around $\Gamma$, the authors implement an infinite-dimensional reduction via a gluing method to couple outer and inner problems. The key innovation is the derivation of a Jacobi–Toda type reduced system governing spike positions and amplitudes, enabling the existence of solutions with $N$ spikes clustering near $\Gamma$ and at mutual distances $O(\varepsilon|\log\varepsilon|)$ as $\varepsilon\to 0$. The results illuminate how curve-concentration phenomena arise in resonant Ambrosetti–Prodi problems and reveal the influence of the matrix $A(x)$ on high-dimensional concentration patterns, with potential applications to anisotropic media and geometric singular perturbation problems.
Abstract
We consider the following Ambrosetti-Prodi type problem \begin{equation}\label{e50} \left\{\begin{array}{ll} -div (A(x)\nabla u)=|u|^p-t\mathbfΨ(x), &\mbox{in $Ω$,} \\ u=0, & \mbox{on $\partial Ω$}, \end{array} \right. \end{equation} where $Ω\subset \mathbb{R}^2$, $t>0$, $p>3$ and $\mathbfΨ$ is an eigenfunction corresponding to the first eigenvalue of the following operator \[\mathfrak{L}(u)=-div (A(x)\nabla u).\] Moreover, $A(x)=\{A_{ij}(x)\}_{2\times 2}$ is a symmetric positive defined matrix function. Let $Γ\subset Ω$ be a closed curve and also a non-degenerate critical point of the functional \[\mathcal{K}(Γ)=\int_Γ\mathbfΨ^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},\] where $\mathfrak{g}(X,Y)=\langle A^*X,Y\rangle$ is a Riemannian metric on $\mathbb{R}^2$ and $A^*$ is the adjoint matrix for $A$. We prove that there exists a sequence of $t=t_l\to +\infty$ such that this problem has solutions $u_{t_l}$ with clustering concentration layers directed along $Γ$.