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 ϵ ps (- Δ) p s u + ϵ qs (- Δ) q s u + V (x) (| u | p - 2 u + | u | q - 2 u) = W (x) g (u) , in R N , u ∈ W s , p (R N) ∩ W s , q (R N) , u > 0 , in R N , <graphic href="12220_2022_983_Article_Equ80.gif"></graphic> where s ∈ (0 , 1) , 2 ≤ p < q < N s , (- Δ) t s , with t ∈ { p , q } , is the fractional t-Laplacian operator, ϵ > 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 ϵ > 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. [ABSTRACT FROM AUTHOR]