MULTIPLICITY OF WEAK SOLUTIONS FOR A (P(X), Q(X))-KIRCHHOFF EQUATION WITH NEUMANN BOUNDARY CONDITIONS

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The aim of this study is to investigate the existence of infinitely many weak solutions for the (p(x), q(x))-Kirchhoff Neumann problem described by the following equation: $ \left\{\begin{array}{lc}-\left(a_1+a_2 \int_{\Omega} \frac{1}{p(x)}|\nabla u|^{p(x)} d x\right) \Delta_{p(\cdot)} u & \\ -\left(b_1+b_2 \int_{\Omega} \frac{1}{q(x)}|\nabla u|^{q(x)} d x\right) \Delta_{q(\cdot)} u & \\ +\lambda(x)\left(|u|^{p(x)-2} u+|u|^{q(x)-2} u\right)=f_1(x, u)+f_2(x, u) & \text { in } \Omega, \\ \frac{\partial u}{\partial \nu}=0 & \text { on } \partial \Omega .\end{array}\right. $ By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.