Multiplicity results involving p-biharmonic Kirchhoff-type problems

• Main Author: Ramzi Alsaedi
• Format: Article
• Language: English
• Published: SpringerOpen 2020-07-01
• Series: Boundary Value Problems
• Subjects: Variational method; Biharmonic Kirchhoff-type equations; Multiple solutions; Nehari manifold
• Online Access: http://link.springer.com/article/10.1186/s13661-020-01416-2

Abstract This paper deals with the existence of multiple solutions for the following Kirchhoff type equations involving p-biharmonic operator: − M ( ∫ Ω ( | Δ p u | 2 + | u | p ) d x ) ( Δ p 2 u − | u | p − 2 u ) = λ f ( x ) | u | q − 2 u + g ( x ) | u | m − 2 u , x ∈ Ω , $$\begin{aligned}& -M \biggl( \int_{\varOmega} \bigl( \vert \Delta_{p}u \vert ^{2}+ \vert u \vert ^{p} \bigr)\,dx \biggr) \bigl( \Delta _{p}^{2}u- \vert u \vert ^{p-2}u \bigr) =\lambda f(x) \vert u \vert ^{q-2}u+g(x) \vert u \vert ^{m-2}u,\quad x\in\varOmega, \end{aligned}$$ where Ω is a bounded domain in R N $\mathbb{R}^{N}$ ( N > 1 $N>1$ ), λ > 0 $\lambda >0$ , p , q , m > 1 $p, q, m>1$ , M is a continuous function, and the weight functions f and g are measurable. We obtain the existence results by combining the variational method with Nehari manifold and fibering maps.