Existence results for a fractional Schrödinger–Poisson equation with concave–convex nonlinearity in ℝ3. (19th October 2021)

• Record Type: Journal Article
• Authors: Jiang, Ruiting; Guo, Fangping; Zhai, Chengbo
• Is Part Of: Mathematical methods in the applied sciences. Volume 45:Number 3(2022)
• Page Start: 1752
• Page End: 1766
• Publication Date: 2021-10-19
• Subjects: critical point theorem -- fractional Schrödinger–Poisson -- general growth conditions -- variational methods; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.7887
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we consider the following nonhomogeneous fractional Schrödinger–Poisson equations: ( − Δ ) s u + V ( x ) u + ϕ u = λ f ( x, u ) + g ( x, u ) in ℝ 3, ( − Δ ) t ϕ = u 2 in ℝ 3, where λ ≥ 0, s ∈ (3/4, 1], t ∈ (0, 1], (− Δ) s denotes the fractional Laplacian. g ( x, u ) is of general 3‐superlinear growth at infinity, f ( x, u ) satisfies sublinear growth conditions. By means of some critical point theorems, we obtain the following results: (1) the existence of at least one solution for λ = 0, (2) the existence of at least two nontrivial solutions for λ > 0 small enough, (3) the infinitely many solutions when f ( x, u ) is odd in u and λ > 0.