Existence and concentration of ground-states for fractional Choquard equation with indefinite potential

Abstract

Abstract This paper is concerned with existence and concentration properties of ground-state solutions to the following fractional Choquard equation with indefinite potential: ( − Δ ) s u + V ( x ) u = ∫ R N A ( ε y ) ∣ u ( y ) ∣ p ∣ x − y ∣ μ d y A ( ε x ) ∣ u ( x ) ∣ p − 2 u ( x ) , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{A\left(\varepsilon y)| u(y){| }^{p}}{| x-y{| }^{\mu }}{\rm{d}}y\right)A\left(\varepsilon x)| u\left(x){| }^{p-2}u\left(x),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where s ∈ ( 0 , 1 ) s\in \left(0,1) , N > 2 s N\gt 2s , 0 < μ < 2 s 0\lt \mu \lt 2s , 2 < p < 2 N − 2 μ N − 2 s 2\lt p\lt \frac{2N-2\mu }{N-2s} , and ε \varepsilon is a positive parameter. Under some natural hypotheses on the potentials V V and A A , using the generalized Nehari manifold method, we obtain the existence of ground-state solutions. Moreover, we investigate the concentration behavior of ground-state solutions that concentrate at global maximum points of A A as ε → 0 \varepsilon \to 0 .