Sign-Changing Solutions for the One-Dimensional Non-Local sinh-Poisson Equation

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.

Keywords

• fractional laplacian
• exponential non-linearities
• non-local
• corrosion modelling
• lyapunov–schmidt reduction
• one-dimension
• sign-changing
• 35r11
• 35j61
• 35b44
• 35b38
• 35b40