Fractional double-phase patterns: concentration and multiplicity of solutions

We consider the following class of fractional problems with unbalanced growth: {(-Delta)(p)(s)u + (-Delta)(q)(s)u + V(epsilon x) (vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(q-2)u) = f(u) in RN, u is an element of W-s,W-p (R-N) boolean AND W-s,W-q (R-N), u < 0 in R-N, where epsilon > 0 is a small parameter, s is an element of (0,1), 2 <= p < q < N/s, (-Delta)(t)(s) (with t is an element of {p,q}) is the fractional t-Laplacian operator, V : R-N -> R is a continuous potential satisfying local conditions, and f : R -> R is a continuous nonlinearity with subcritical growth. Applying suitable variational and topological arguments, we obtain multiple positive solutions for epsilon > 0 sufficiently small as well as related concentration properties, in relationship with the set where the potential V attains its minimum. (C) 2020 Elsevier Masson SAS. All rights reserved.