Existence and multiplicity of solutions for nonlocal fourth-order p ( x )-Kirchhoff problems with Leray--Lions operators, Hardy potentials, and logarithmic nonlinearities

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Abstract This paper is devoted to the existence and multiplicity of weak solutions for a class of nonlocal fourth-order p ⁢ ( x ) {p(x)} -Kirchhoff problems involving a Leray–Lions type operator, Hardy-type singular potentials, and logarithmic nonlinear reaction terms in the framework of variable exponent Sobolev spaces. More precisely, we investigate the boundary value problem geändert { M ⁢ ( ∫ Ω A ⁢ ( x , Δ ⁢ u ) ⁢ 𝑑 x ) ⁢ Δ ⁢ ( a ⁢ ( x , Δ ⁢ u ) ) + θ ⁢ ( x ) ⁢ | u | s ⁢ ( x ) - 2 ⁢ u | x | 2 ⁢ s ⁢ ( x ) + c ⁢ ( x ) ⁢ | u | q ⁢ ( x ) - 2 ⁢ u = λ ⁢ f ⁢ ( x , u ) + μ ⁢ g ⁢ ( x , u ) + β ⁢ | u | r ⁢ ( x ) - 2 ⁢ u ⁢ ln ⁡ ( 1 + | u | ) , in ⁢ Ω , u = Δ ⁢ u = 0 , on ⁢ ∂ ⁡ Ω , see text \left\{\begin{aligned} &\displaystyle M\Biggl{(}\int_{\Omega}A(x,\Delta u)\,dx% \Biggr{)}\Delta(a(x,\Delta u))+\frac{\theta(x)|u|^{s(x)-2}u}{|x|^{2s(x)}}+c(x)% |u|^{q(x)-2}u\\ &\displaystyle=\lambda f(x,u)+\mu g(x,u)+\beta|u|^{r(x)-2}u\ln(1+|u|),&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle\quad u&\displaystyle=\Delta u=0,&&\displaystyle\phantom{}\text{% on }\partial\Omega,\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ) is a bounded domain with smooth boundary, M is a nonlocal Kirchhoff function, a ⁢ ( x , ⋅ ) {a(x,\cdot\,)} is a Leray–Lions type operator satisfying suitable structural assumptions, and the singular weighted term | u | s ⁢ ( x ) - 2 ⁢ u | x | 2 ⁢ s ⁢ ( x ) {\frac{|u|^{s(x)-2}u}{|x|^{2s(x)}}} represents a Hardy-type potential. The additional logarithmic perturbation β ⁢ | u | r ⁢ ( x ) - 2 ⁢ u ⁢ ln ⁡ ( 1 + | u | ) {\beta|u|^{r(x)-2}u\ln(1+|u|)} models a broad class of nonlinear reaction mechanisms exhibiting intermediate growth between polynomial and super-polynomial nonlinearities. The nonlinearities f and g satisfy suitable Carathéodory conditions, while λ > 0 {\lambda>0} , μ ≥ 0 {\mu\geq 0} , and β ≥ 0 {\beta\geq 0} are real parameters. Using variational methods, compact embedding theorems in variable exponent Sobolev spaces, Hardy–Rellich inequalities, monotonicity techniques associated with Leray–Lions operators, and the Three Critical Points Theorems of Bonanno–Marano and Bonanno–Candito, we establish the existence of at least three distinct weak solutions. The presence of the logarithmic reaction term requires a refined variational analysis owing to its nonhomogeneous growth and the delicate compactness properties arising in variable exponent settings. The obtained results significantly generalize and unify several existing multiplicity theorems for fourth-order p ⁢ ( x ) {p(x)} -Kirchhoff and p ⁢ ( x ) {p(x)} -biharmonic problems by simultaneously incorporating nonlocal Kirchhoff effects, Leray–Lions structures, Hardy-type singular potentials, variable exponent logarithmic nonlinearities, and additional nonlinear perturbations within a unified variational framework. To the best of our knowledge, this paper provides one of the first multiplicity results for nonlocal fourth-order variable exponent Kirchhoff problems combining all these features in a single mathematical model.