Nontrivial Solutions and Asymptotic Behavior of Variable Critical p(x)-Laplace Equations with Negative Nonlocal Kirchhoff Terms

We study a Dirichlet problem for a p(x)-Kirchhoff equation with negative nonlocal structure and variable critical nonlinearity. The modified Kirchhoff term and variable critical exponent break Sobolev compactness and complicate Palais–Smale sequence analysis. Using a truncation argument, the mountain pass theorem and the variable exponent concentration compactness principle, we prove the existence of a nontrivial weak solution for sufficiently large λ, with the solution norm decaying to zero as λ→+∞. This extends subcritical results to the variable critical framework.