Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

Abstract

Abstract We study existence of solutions for the fractional problem ( P m ) ( − Δ ) s u + μ u = g ( u ) in R N , ∫ R N u 2 d x = m , u ∈ H r s ( R N ) , where N ⩾ 2, s ∈ (0, 1), m > 0, μ is an unknown Lagrange multiplier and g ∈ C ( R , R ) satisfies Berestycki–Lions type conditions. Using a Lagrangian formulation of the problem (P m ), we prove the existence of a weak solution with prescribed mass when g has L 2 subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev’s mountain in a product space and some deformation arguments under a new version of the Palais–Smale condition introduced in Hirata and Tanaka (2019 Adv. Nonlinear Stud. 19 263–90); Ikoma and Tanaka (2019 Adv. Differ. Equ. 24 609–46). A multiplicity result of infinitely many normalized solutions is also obtained if g is odd.