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

TL;DR

This paper studies solutions to a nonlinear PDE with concentration layers along a curve, extending the understanding of clustering phenomena in Ambrosetti-Prodi type problems with variable coefficients.

Contribution

It introduces a new method using clustering concentration layers along a critical curve for solutions of the Ambrosetti-Prodi problem with variable coefficients.

Findings

Existence of solutions with clustering layers along a specified curve.

Solutions form as the parameter t approaches infinity.

The approach applies to problems with variable coefficient matrices.

Abstract

We consider the following Ambrosetti-Prodi type problem \begin{equation} \left\{\begin{array}{ll} -\mathrm{div} (A(x)\nabla u)=|u|^p-t\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\ u=0, & \mbox{on $\partial \Omega$}, \end{array} \right. \end{equation} where $\Omega \subset \mathbb{R}^2$, $t>0$, $p>3$ and $\mathbf{\Psi}$ is an eigenfunction corresponding to the first eigenvalue of the following operator \[\mathfrak{L}(u)=-\mathrm{div} (A(x)\nabla u).\] Moreover, $A(x)=\{A_{ij}(x)\}_{2\times 2}$ is a symmetric positive defined matrix function. Let $\Gamma \subset \Omega$ be a closed curve and also a non-degenerate critical point of the functional \[\mathcal{K}(\Gamma)=\int_\Gamma \mathbf{\Psi}^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},\] 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 $\Gamma$.