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.

Automated summary

This paper deals with a singularly perturbed fractional double-phase problem with unbalanced growth and competing potentials. By utilizing topological and variational methods, the existence and concentration phenomena of positive solutions are established, as well as the multiplicity result dependent on the topology of the set where the potentials attain their extrema.