On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory - Taarabti, S.; El Allali , Z.; Ben Haddouch, K. (Ukrains’kyi Matematychnyi Zhurnal, 2020)
UDC 517.9The paper deals with the existence and multiplicity of nontrivial weak solutions for the $p(x)$-Kirchhoff-type problem$$ {-M}\!\left(\displaystyle\int\limits_{Ω}\frac{1}{p(x)}|Δ u|^{p(x)}\,dx\right)\!Δ_{p(x)}^{2} u = f(x,u)\quad in\quad Ω, $$$$ u = Δ u = 0\quad on\quad \partialΩ.$$By using variational approach and Krasnoselskii's genus theory, we prove the existence and multiplicity of solutions for the $p(x)$-Kirchhoff-type equation.