On multiplicity of positive solutions for nonlocal equations with critical nonlinearity. (August 2020)

• Record Type: Journal Article
• Authors: Bhakta, Mousomi; Pucci, Patrizia
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: 35R11 -- 35A15 -- 35B33 -- 35J60; Nonlocal equations -- Fractional Laplacian -- Palais–Smale decomposition -- Energy estimate -- Positive solutions -- Min–max method; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111853
• Languages: English
• ISSNs: 0362-546X

Abstract

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 ).