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

Abstract

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.

Keywords

• Singular elliptic problem
• Variational methods
• Palais–Smale condition