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 $\hat{I}\text{c◯}$,} \\ u=0, & \mbox{on $\partial \hat{I}\text{c◯}$}, \end{array} \right. \end{equation} where $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$, $t>0$, $p>3$ and $\hat{\mathbf{I}}¨$ 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 $\hat{I}“\subset \hat{I}\text{c◯}$ 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 $\mathfrak{g}(X,Y)=⟨A^{\ast }X,Y⟩$ 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 $t=t_l\to +\infty$ such that this problem has solutions $u_{t_l}$ with clustering concentration layers directed along $\hat{I}“$.

57 pages