In this paper, we consider a class of nonlocal boundary value problems driven by the \(p(x)\) -triharmonic operator and involving nonlocal integral terms under Neumann boundary conditions. The analysis is carried out within the framework of variable exponent Sobolev spaces. By applying Ekeland’s variational principle, we establish the existence of at least one weak solution. Moreover, under appropriate symmetry assumptions, we prove the existence of infinitely many solutions. We also derive a regularity result showing that the weak solutions possess higher smoothness than initially assumed. Additionally, we establish a stability (or bifurcation) result illustrating how the solutions respond to variations in parameters. Collectively, these findings provide a comprehensive and in-depth understanding of the problem, thereby enhancing the mathematical and physical relevance of the study.
