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

We consider the following Ambrosetti-Prodi type problem \begin{equation} \left{\begin{array}{ll} -$\mathrm{d}\mathrm{i}\mathrm{v}$ (A(x)\nabla u)=|u|^p-t\mathbfΨ(x), &\mbox{in $\mathrm{\Omega }$,} \ u=0, & \mbox{on $\mathrm{\partial }\mathrm{\Omega }$}, \end{array} \right. \end{equation} where $\mathrm{\Omega }\subset {\mathbb{R}}^2$, $t>0$, $p>3$ and $\mathbf{\Psi }$ is an eigenfunction corresponding to the first eigenvalue of the following operator $$\mathfrak{L}(u)=-\mathrm{d}\mathrm{i}\mathrm{v}(A(x)\mathrm{\nabla }u)\mathrm{.}$$ Moreover, $A(x)=\{A_{ij}(x){\}}_{2\times 2}$ is a symmetric positive defined matrix function. Let $\mathrm{\Gamma }\subset \mathrm{\Omega }$ be a closed curve and also a non-degenerate critical point of the functional $$\mathcal{K}(\mathrm{\Gamma })={\int }_{\mathrm{\Gamma }}{\mathbf{\Psi }}^{\frac{p+3}{2p}}dvo{l}_{\mathfrak{g}},$$ where $\mathfrak{g}(X,Y)=⟨A^{\ast }X,Y⟩$ is a Riemannian metric on ${\mathbb{R}}^2$ and $A^{\ast }$ is the adjoint matrix for $A$. We prove that there exists a sequence of $t=t_l\to +\mathrm{\infty }$ such that this problem has solutions $u_{t_l}$ with clustering concentration layers directed along $\mathrm{\Gamma }$.