Existence of Solutions for a Fractional Relativistic Schrödinger Equation with Indefinite Potentials

This paper is devoted to the following fractional relativistic Schrödinger equation:

$$(-\Delta+m^{2})^{s}u+V(x)u=f(x,u),\qquad x\in\mathbb{R}^{N},$$

where (−Δ + m²)^s is the fractional relativistic Schrödinger operator, s ∈ (0, 1), m > 0, V: ℝ^N → ℝ is a continuous potential and f: ℝ^N × ℝ → ℝ is a superlinear continuous nonlinearity with subcritical growth. We consider the case where the potential V is indefinite so that the relativistic Schrödinger operator (−Δ + m²)^s + V possesses a finite-dimensional negative space. With the help of extension method and Morse theory, the existence of a nontrivial solution for the above problem is obtained.