Bound state for fractional Schrödinger equation with saturable nonlinearity

摘要

In this paper, we study the existence of bound state for the following fractional Schrödinger equation (P)(−Δ)αu+V(x)u=f(u),x∈RN,N≥3,where (−Δ)α with α ∈ (0, 1) is the fractional Laplace operator defined as a pseudo-differential operator with the symbol |ξ|2α, V(x) is a positive potential function and the nonlinearity f is saturable, that is, f(u)/u→l∈(0,+∞) as |u|→+∞. By using a variant version of Mountain Pass Theorem, we prove that there exists a bound state and ground state of (P) when V and f satisfy suitable assumptions.