We study existence of solutions for the fractional problem (Pm) ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ (−Δ)su + μu = g(u) inRN, RN u2dx = m, u ∈ Hs r (RN), 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 (Pm), we prove the existence of a weak solution with prescribed mass when g has L2 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.
Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
CINGOLANI S
;GALLO M;
2021-01-01
Abstract
We study existence of solutions for the fractional problem (Pm) ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ (−Δ)su + μu = g(u) inRN, RN u2dx = m, u ∈ Hs r (RN), 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 (Pm), we prove the existence of a weak solution with prescribed mass when g has L2 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
