The nonlinear fractional relativistic schrodinger equation: Existence, multiplicity, decay and concentration results

In this paper we study the following class of fractional relativistic Schrödinger equations: {eqution presented} where " > 0 is a small parameter, s 2 (0; 1), m > 0, N > 2s, (-Δ + m2)s is the fractional relativistic Schrödinger operator, V: RN→ R is a continuous potential satisfying a local condition, and f: R → R is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for ϵ > 0 small enough, the above problem admits a weak solution u" which concentrates around a local minimum point of V as ϵ → 0. We also show that uϵ has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential V attains its minimum value.