Existence of infinitely many solutions for a p-Kirchhoff problem in RN

• Main Authors: Zonghu Xiu, Jing Zhao, Jianyi Chen
• Format: Article
• Language: English
• Published: SpringerOpen 2020-06-01
• Series: Boundary Value Problems
• Subjects: Singular elliptic problem; Variational methods; Palais–Smale condition
• Online Access: http://link.springer.com/article/10.1186/s13661-020-01403-7

Abstract We consider the existence of multiple solutions of the following singular nonlocal elliptic problem: { − M ( ∫ R N | x | − a p | ∇ u | p ) div ( | x | − a p | ∇ u | p − 2 ∇ u ) = h ( x ) | u | r − 2 u + H ( x ) | u | q − 2 u , u ( x ) → 0 as  | x | → ∞ , $$\begin{aligned} \textstyle\begin{cases} -M(\int _{\mathbb{R} ^{N}}{ \vert x \vert ^{-ap} \vert \nabla u \vert ^{p}})\operatorname{div}( \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\nabla u)= h(x) \vert u \vert ^{r-2}u+H(x) \vert u \vert ^{q-2}u, \\ u(x)\rightarrow 0 \quad \text{as } \vert x \vert \rightarrow \infty , \end{cases}\displaystyle \end{aligned}$$ where x ∈ R N $x\in \mathbb{R} ^{N}$ , and M ( t ) = α + β t $M(t)=\alpha +\beta t$ . By the variational method we prove that the problem has infinitely many solutions when some conditions are fulfilled.