Solutions to Nonlocal p-Laplacian Problems with Critical Growth in Complex Materials"

What is it about?

This paper studies a class of nonlinear, nonlocal equations known as p-Laplacian Kirchhoff problems, which model materials or systems whose stiffness changes depending on their overall deformation—a feature important in elastic strings, metamaterials, and similar structures. We tackle the hardest case, where the system has critical growth (making standard solution methods fail) and where key coefficients can weaken or vanish in parts of the domain. Using advanced variational techniques and the mountain pass theorem, we prove when such systems have physically meaningful solutions, describe their energy properties, and show how these solutions disappear or concentrate as parameters reach critical thresholds. We also give practical examples of material profiles that fit the theory, making the results applicable to real-world engineering and physics problems.

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