MULTIPLE SOLUTIONS FOR NONLOCAL FOURTH-ORDER VARIABLE EXPONENT KIRCHHOFF PROBLEMS WITH WEIGHTED HARDY–RELLICH POTENTIALS AND NONLINEAR ROBIN BOUNDARY CONDITIONS

In this paper, we investigate the existence and multiplicity of weak solutions for a class of nonlocal fourth-order Leray–Lions–Kirchhoff problems with variable exponent growth, weighted Hardy–Rellich singular potentials with logarithmic corrections, and nonlinear Robin boundary conditions. The model couples a generalized fourth-order Leray–Lions operator with a Kirchhoff-type nonlocal mechanism and singular variable-exponent effects, leading to a variational structure in which higher-order deformation, weighted singularity, and nonlinear boundary interactions must be controlled simultaneously. Working in variable exponent Sobolev and trace spaces, we establish the well-definedness, coercivity, sequential weak lower semicontinuity, and continuous Fréchet differentiability of the principal functional, together with the compactness properties of the nonlinear perturbations. The analysis combines weighted Hardy–Rellich inequalities, variable exponent Sobolev and trace embeddings, modular estimates, and monotonicity properties of Leray–Lions operators. By applying three-critical-points principles of Bonanno–Candito and Bonanno–Marano, we prove the existence of at least three distinct weak solutions for suitable ranges of the parameters. The multiplicity mechanism is obtained through explicit geometric estimates on low-energy sublevels and a localized comparison function adapted to the fourth-order variational structure. The results extend existing multiplicity theory by treating generalized fourth-order Leray–Lions diffusion, Kirchhoff nonlocality, variable exponent growth, weighted Hardy–Rellich logarithmic singularities, and nonlinear boundary interactions simultaneously within a unified critical-point framework.