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

[摘要] This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction: {(-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) = h(u) + vertical bar u vertical bar(r-2)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 in R-N,} where s is a positive parameter, 0 < s < 1, 2 <= p q(s)*. 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 epsilon is small enough and in close relationship with the set where the potential V attains its minimum. (C) 2021 The Author(s). Published by Elsevier Inc.

[发布日期] 2021-11-25

[关键词] Fractional double-phase problem;Double phase energy;Nehari manifold;Penalization;Critical problem;Supercritical problem