Existence of positive solutions for nonlocal problems with indefinite nonlinearity

Abstract

Abstract In this paper, we consider the following new nonlocal problem: { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ f ( x ) | u | p − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , $$\left \{ \textstyle\begin{array}{l@{\quad}l} - (a-b\int_{\varOmega} \vert \nabla u \vert ^{2}\,dx )\Delta u=\lambda f(x) \vert u \vert ^{p-2}u, & x\in\varOmega,\\u=0, & x\in\partial\varOmega, \end{array}\displaystyle \right . $$ where Ω is a smooth bounded domain in R 3 $\mathbb{R}^{3}$ , a , b > 0 $a,b>0$ are constants, 3 0 $\lambda>0$ . Under some assumptions on the sign-changing function f, we obtain the existence of positive solutions via variational methods.

Keywords

• Nonlocal problem
• Variational methods
• Positive solution
• Indefinite nonlinearity