On multiplicity of positive solutions for nonlocal equations with critical nonlinearity. (August 2020) - Bhakta, Mousomi; Pucci, Patrizia
Abstract: This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: ( E ) ( − Δ ) s u = a ( x ) | u | 2 s ∗ − 2 u + f ( x ) in  R N, u ∈ H ̇ s ( R N ), where s ∈ ( 0, 1 ), N > 2 s, 2 s ∗ ≔ 2 N N − 2 s, 0 < a ∈ L ∞ ( R N ) and f is a nonnegative nontrivial functional in the dual space of H ̇ s ( R N ) i.e., ( H ̇ s ) ′ 〈 f, u 〉 H ̇ s ≥ 0, whenever u is a nonnegative function in H ̇ 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 | x | → ∞ and ‖ f ‖ H ̇ s ( R N ) ′ is small enough (but f ⁄ ≡ 0 ), we establish existence of at least two positive solutions to ( E ).