Multiple solutions for fractional p-Laplace equation with concave-convex nonlinearities

Abstract

Abstract In this paper, we investigate the existence of solutions for the fractional p-Laplace equation ( − Δ ) p s u + V ( x ) | u | p − 2 u = h 1 ( x ) | u | q − 2 u + h 2 ( x ) | u | r − 2 u in R N , $$ (-\Delta)_{p}^{s}u+V(x) \vert u \vert ^{p-2}u=h_{1}(x) \vert u \vert ^{q-2}u+h_{2}(x) \vert u \vert ^{r-2}u \quad \mbox{in } \mathbb{R}^{N}, $$ where N > s p $N>sp$ , 0 0 $V(x)>0$ and h 1 ( x ) $h_{1}(x)$ , h 2 ( x ) $h_{2}(x)$ are allowed to change sign in R N $\mathbb {R}^{N}$ . By using variant fountain theorem, we prove that the above equation admits infinitely many small and high energy solutions.

Keywords

• Fractional p-Laplacian equations
• Variational methods
• Variant fountain theorem