Multiple solutions for a nonlinear p ( x )-curl system with nonlocal terms via a three critical points theorem

This paper addresses the existence of multiple weak solutions for a nonlinear curl-type system involving variable exponent growth and nonlocal source terms. We study the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {p(x)} -curl problem posed in a bounded, simply connected domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝔻</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> {D³} , subject to divergence-free and mixed boundary conditions. The system features spatially variable nonlinearities and two integral-type nonlocal terms governed by Carathéodory functions with subcritical growth. Utilizing the framework of variable exponent Sobolev spaces and a three critical points theorem established by Bonanno and Morano, we prove the existence of at least three weak solutions under suitable structural assumptions. This work extends recent developments on variable exponent curl systems and offers new insights into the mathematical modeling of complex physical phenomena exhibiting spatial heterogeneity and nonlocal interactions.