The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data

Abstract

Let \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document} be a bounded smooth domain, we study the following anisotropic elliptic problem \begin{document}$ \begin{cases} -\nabla(a(x)\nabla \upsilon)+a(x)\upsilon = 0& \text{in}\, \, \, \, \, \Omega, \dfrac{\partial \upsilon}{\partial\nu} = e^\upsilon-s\phi_1-h(x) & \text{on}\, \, \partial\Omega, \end{cases} $\end{document} where \begin{document}$ \nu $\end{document} denotes the outer unit normal vector to \begin{document}$ \partial\Omega $\end{document} , \begin{document}$ h\in C^{0, \alpha}( \partial\Omega) $\end{document} , \begin{document}$ s>0 $\end{document} is a large parameter, \begin{document}$ a(x) $\end{document} is a positive smooth function and \begin{document}$ \phi_1 $\end{document} is a positive first Steklov eigenfunction. We show that this problem has an unbounded number of solutions for all sufficiently large \begin{document}$ s $\end{document} , which give a positive answer to a generalization of the Lazer-McKenna conjecture for this case. Moreover, the solutions found exhibit multiple concentration behavior around boundary maxima of \begin{document}$ a(x)\phi_1 $\end{document} as \begin{document}$ s\rightarrow+\infty $\end{document} .