Multiplicity and Concentration of Positive Solutions for Fractional Unbalanced Double-Phase Problems

This paper is concerned with the following singularly perturbed fractional double-phase problem with unbalanced growth and competing potentials {epsilon(ps) (-Delta)(p)(s)u + epsilon(qs) (-Delta)(q)(s) u + V(x) (vertical bar u vertical bar(p-2)u + vertical bar u vertical bar(q-2)u) = W(x)g(u), in R-N, u is an element of W-s,W- p (R-N) boolean AND W-s,W- q(R-N), u > 0, where 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, epsilon > 0 is a small parameter, V is the absorption potential, W is the reaction potential and g is the reaction term with subcritical growth. Assume that the potentials V, W, and the nonlinearity g satisfy some natural conditions, applying topological and variational methods, we establish the existence and concentration phenomena of positive solutions for epsilon > 0 sufficiently small as well as the multiplicity result depended on the topology of the set where V attains its global minimum and W attains its global maximum. Finally, we also obtain the nonexistence result of ground state solutions under suitable conditions.