By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equationwhere ε > 0 is a parameter, s ∈ (0, 1), N > 2s, (−∆) s is the fractional Laplacian, V is a positive continuous potential with local minimum, and f is a superlinear function with subcritical growth. We also obtain a multiplicity result when f (u) = |u| q−2 u + λ|u| r−2 u with 2 < q < 2 * s ≤ r and λ > 0, by combining a truncation argument and a Moser-type iteration.2010 Mathematics Subject Classification. 35A15, 35J60, 35R11, 45G05.

No abstract

Abstract. In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic fieldwhere ε > 0 is a parameter, s ∈ (0, 1), N ≥ 3, (−∆) s A is the fractional magnetic Laplacian, V : R N → R and A : R N → R N are continuous potentials and f : R N → R is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for ε small.

Abstract. In this work we study the following fractional scalar field equations is the fractional Laplacian and the nonlinearity g ∈ C 2 (R) is such that g ′′ (0) = 0. By using variational methods, we prove the existence of a positive solution which is spherically symmetric and decreasing in r = |x|.

No abstract