Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations

We are concerned with the following p-biharmonic equations: $$ \Delta _{p}^{2} u+M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr) \operatorname{div}\bigl(\varphi (x,\nabla u)\bigr)+V(x) \vert u \vert ^{p-2}u=\lambda f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ where \(2< 2pDelta _{p}^{2}u=\Delta (|\Delta u|^{p-2} \Delta u)\), the function \(\varphi (x,v)\) is of type \(\lvert v \rvert ^{p-2}v\), \(\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)\), the potential function \(V:\mathbb{R}^{N}\to (0,\infty )\) is continuous, and \(f:\mathbb{R} ^{N}\times \mathbb{R} \to \mathbb{R}\) satisfies the Caratheodory condition. We study the existence of weak solutions for the problem above via mountain pass and fountain theorems.