High perturbations of a new Kirchhoff problem involving the p-Laplace operator

Abstract

Abstract In the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ where a , b > 0 $a, b > 0$ , Δ p u : = div ( | ∇ u | p − 2 ∇ u ) $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the p-Laplace operator, 1 < p < N $1 < p < N$ , p < q < p ∗ : = ( N p ) / ( N − p ) $p < q < p^{\ast }:=(Np)/(N-p)$ , Ω ⊂ R N $\Omega \subset \mathbb{R}^{N}$ ( N ≥ 3 $N \geq 3$ ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.

Keywords

• p-Laplace operator
• New Kirchhoff problem
• Variational method