Sign-changing solutions for the one-dimensional non-local sinh-Poisson equation

TL;DR

This paper constructs sign-changing solutions for a non-local sinh-Poisson equation on a 1D interval, revealing the structure of solutions with multiple alternating peaks related to electrochemical corrosion models.

Contribution

It introduces a finite-dimensional reduction approach to explicitly construct solutions with prescribed sign-changing peaks for the non-local sinh-Poisson equation.

Findings

Solutions with arbitrarily many sign-changing peaks are constructed.

The number of nodal regions matches the number of blow-up points.

The solutions model galvanic corrosion phenomena.

Abstract

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval $I$, under Dirichlet conditions in the exterior of $I$. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov-Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.