The Gel’fand Problem on Expanding Tubular Domains in \(\mathbb {R}^2\): Existence and the Morse Index of Solutions

We discuss the existence and the Morse index of solutions to the problem \(\begin{aligned} {\left\{ \begin{array}{ll} \Delta U+\lambda e^U=0 & \text {in}\ \Omega _{R},\\ U=0 & \text {on}\ \partial \Omega _{R}, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }U+\lambda e^U=0\hfill & \text{in}{\mathrm{\Omega }}_R,\hfill \\ U=0\hfill & \text{on}\partial {\mathrm{\Omega }}_R,\hfill \end{array}\right)\end{array}$$ where \(\Omega _R\) ${\mathrm{\Omega }}_R$ is a tubular domain in the plane with fixed width. We obtain the existence of an increasing sequence \(R_k\rightarrow \infty \) $R_k\to \infty$ as \(k\rightarrow \infty \) $k\to \infty$ and a corresponding sequence of solutions \(\{U_{R_k}\}_{k\ge 1}\) ${\{U_{R_k}\}}_{k\ge 1}$ to the above problem. We investigate the energy of such solutions and show that their Morse index \({{\textsf{m}}}(U_{R_k})\) $\mathsf{m}(U_{R_k})$ satisfies \({{\textsf{m}}}(U_{R_k})/R_k\rightarrow 2\sqrt{-\eta _1}\) $\mathsf{m}(U_{R_k})/R_k\to 2\sqrt{-{\eta }_1}$ as \(k\rightarrow \infty \) $k\to \infty$ , where \(\eta _1<0\) ${\eta }_1<0$ is the first eigenvalue of the linearized 1D Gel’fand problem.