Multiple solutions for a class of \(p(x)\)-Kirchhoff type problems with Neumann boundary conditions (Q2841517)

The paper is concerned with the \(p(x)\)-Kirchhoff Neumann problem NEWLINE\[NEWLINE \begin{cases} \displaystyle -M\left(\int_\Omega \frac1{p(x)}|\nabla u|^{p(x)} dx \right)\text{div}\big(|\nabla u|^{p(x)-2}\nabla u \big) =f(x,u) & \text{ in } \Omega,\cr \displaystyle \frac{\partial u}{\partial \nu}=0 & \text{ on } \partial\Omega \end{cases} NEWLINE\]NEWLINE with \(p(x)\in C(\bar\Omega).\) It is proved a multiplicity result for the above problem using an abstract linking argument due to Brézis and Nirenberg.