Exact solution structures on some nonlocal overdetermined problems

TL;DR

This paper investigates the solution structures of nonlocal overdetermined problems of Serrin type with Kirchhoff terms, establishing the exact number of solutions and deriving explicit forms using solutions from local problems.

Contribution

It provides a novel analysis of nonlocal overdetermined problems, linking solution counts to transcendental equations and deriving explicit solutions.

Findings

Number of solutions matches solutions of specific transcendental equations

Explicit solutions derived from local overdetermined problems

Solution structures characterized for Kirchhoff-type nonlocal problems

Abstract

In this paper, we study the solution structures of Serrin-type overdetermined problems with Kirchhoff-type nonlocal terms. We prove that the exact number of solutions is the same as those of some transcendental equations defined by the nonlocal terms. We also obtain the explicit form of solutions by using the unique solutions of the overdetermined problems without the nonlocal terms.