Concentration of solutions for fractional double-phase problems: critical and supercritical cases

Abstract

This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction:{(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=h(u)+|u|r−2u in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0, in RN,} where ε is a positive parameter, 0<s<1, 2⩽p<q<N/s, (−Δ)ts(t∈{p,q}) is the fractional t-Laplace operator, while V:RN↦R and h:R↦R are continuous functions. The analysis developed in this paper covers both critical and supercritical cases, that is, we assume that either r=qs⁎:=Nq/(N−sq) or r>qs⁎. The main results establish the existence of multiple positive solutions as well as related concentration properties. In the first case, due to the strong influence of the critical term, the result holds true for “high perturbations” of the subcritical nonlinearity. In the second framework, the result holds true for “low perturbations” of the supercritical nonlinearity. The concentration properties are achieved by combining topological and variational methods, provided that ε is small enough and in close relationship with the set where the potential V attains its minimum.