Solutions with clustering concentration layers to the Ambrosetti-Prodi type problem

Abstract

We consider the following Ambrosetti-Prodi type problem \begin{equation} \left\{\begin{array}{ll} -\mathrm{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 \[\mathfrak{L}(u)=-\mathrm{div} (A(x)\nabla u).\] 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 $Γ$.