Multiple concentrating solutions for a fractional (p, q)-Choquard equation

Abstract

We focus on the following fractional (p, q)-Choquard problem: ( − Δ ) p s u + ( − Δ ) q s u + V ( ε x ) ( | u | p − 2 u + | u | q − 2 u ) = 1 | x | μ * F ( u ) f ( u ) in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0 in R N , $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=\left(\frac{1}{\vert x{\vert }^{\mu }}{\ast}F\left(u\right)\right)f\left(u\right) \,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \\ u\in {W}^{s,p}\left({\mathbb{R}}^{N}\right)\cap {W}^{s,q}\left({\mathbb{R}}^{N}\right), u{ >}0\,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \end{cases}$ where ɛ > 0 is a small parameter, 0 < s < 1, 1 < p < q < N s $1{< }p{< }q{< }\frac{N}{s}$ , 0 < μ < sp, ( − Δ ) r s ${\left(-{\Delta}\right)}_{r}^{s}$ , with r ∈ {p, q}, is the fractional r-Laplacian operator, V : R N → R $V:{\mathbb{R}}^{N}\to \mathbb{R}$ is a positive continuous potential satisfying a local condition, f : R → R $f:\mathbb{R}\to \mathbb{R}$ is a continuous nonlinearity with subcritical growth at infinity and F ( t ) = ∫ 0 t f ( τ ) d τ $F\left(t\right)={\int }_{0}^{t}f\left(\tau \right) \mathrm{d}\tau $ . Applying suitable variational and topological methods, we relate the number of solutions with the topology of the set where the potential V attains its minimum value.