On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: {(-Delta)(s)u= a(x)vertical bar u vertical bar(2)*s(-2) u + f(x) in R-N, (epsilon) u epsilon (H) over dot(s)(R-N), where s epsilon (0, 1), N > 2s, 2*(s) := 2N/N-2s, 0 < a epsilon L infinity(R-N) and f is a nonnegative nontrivial functional in the dual space of (H) over dot(s)(R-N) i.e., ((H) over dot(s))' f, u ((H) over dots) >= 0, whenever u is a nonnegative function in (H) over dot(s)(R-N). We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that a is a continuous function, a(x) >= 1 in R-N, a(x) -> 1 as vertical bar x vertical bar -> 8 and vertical bar vertical bar f vertical bar vertical bar ((H) over dots(RN)') is small enough (but f not equivalent to 0), we establish existence of at least two positive solutions to (epsilon). (C) 2020 Elsevier Ltd. All rights reserved.