Existence, multiplicity and concentration for a class of fractional <tex-math id="M1">$ p & q $</tex-math> Laplacian problems in <tex-math id="M2">R N</tex-math>

In this work we consider the following class of fractional $p & q$ Laplacian problems equation* (-Δ)ₚˢu+ (-Δ)qˢu + V( ε x) (|u|ᵖ⁻²u + |u|q-2u) = f(u) in R N, equation* where ε >0 is a parameter, s∈ (0, 1), 1< p<q<N/s, (-Δ)ˢₜ, with t∈ ,q\, is the fractional t-Laplacian operator, V: R N→ R is a continuous potential and f: R → R is a C ¹-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that ε is sufficiently small.