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

Abstract

This paper is concerned with existence and concentration properties of ground-state solutions to the following fractional Choquard equation with indefinite potential: (−Δ)su+V(x)u=∫RNA(εy)∣u(y)∣p∣x−y∣μdyA(εx)∣u(x)∣p−2u(x),x∈RN,{\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>2sN\gt 2s, 0<μ<2s0\lt \mu \lt 2s, 2<p<2N−2μN−2s2\lt p\lt \frac{2N-2\mu }{N-2s}, and ε\varepsilon is a positive parameter. Under some natural hypotheses on the potentials VV and AA, 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 AA as ε→0\varepsilon \to 0.

Keywords

• fractional choquard equation
• concentration
• ground-state solutions
• 35a15
• 35b09
• 35j92
• 58e05