New multiplicity of positive solutions for some class of nonlocal problems

Abstract

Abstract In this paper, we study the following nonlocal problem: { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ where Ω is a smooth bounded domain in R N $\mathbb{R}^{N}$ with N ≥ 3 $N\ge 3$ , a , b > 0 $a,b>0$ , 1 0 $\lambda >0$ is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as b ↘ 0 $b\searrow 0$ .

Keywords

• Variational method
• Nonlocal problem
• Multiple positive solutions
• Blow up