Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations - Jung-Hyun Bae (Boundary Value Problems, 2019)
Abstract We are concerned with the following p-biharmonic equations: Δp2u+M(∫RNΦ0(x,∇u)dx)div(φ(x,∇u))+V(x)|u|p−2u=λf(x,u)in RN, $$ \Delta _{p}^{2} u+M \biggl( \int _{\mathbb{R}^{N}}\varPhi _{0}(x,\nabla u) \,dx \biggr) \operatorname{div}\bigl(\varphi (x,\nabla u)\bigr)+V(x) \vert u \vert ^{p-2}u=\lambda f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ where 2<2p<N $2< 2p<N$, Δp2u=Δ(|Δu|p−2Δu) $\Delta _{p}^{2}u=\Delta (|\Delta u|^{p-2} \Delta u)$, the function φ(x,v) $\varphi (x,v)$ is of type |v|p−2v $\lvert v \rvert ^{p-2}v$, φ(x,v)=ddvΦ0(x,v) $\varphi (x,v)=\frac{d}{dv}\varPhi _{0}(x,v)$, the potential function V:RN→(0,∞) $V:\mathbb{R}^{N}\to (0,\infty )$ is continuous, and f:RN×R→R $f:\mathbb{R} ^{N}\times \mathbb{R} \to \mathbb{R}$ satisfies the Carathéodory condition. We study the existence of weak solutions for the problem above via mountain pass and fountain theorems.