This paper studies the existence of normalized solutions for nonlinear equations involving mixed fractional Laplacians with different orders: { ( − ∆ ) s 1 u +( − ∆ ) s 2 u + V ( x ) u = λu + | u | p − 2 u , x ∈ R d , ∫ R d | u | 2 dx = c 2 , where 1 ≤ d < 2 s 1 s 2 s 2 − s 1 , 0 < s 1 < s 2 < 1 , 2 < p < 2 + 4 s 2 d and the potential V : R d → [ 0 , + ∞ ) is a bounded and continuous function. This type of equation appears in many fields and has been increasingly attracting the attention of scientists in recent years due to its numerous and important applications. By using variational methods, we can prove that, when V satisfies four different assumptions, the minimization of the energy functional is achieved.

In this paper, we consider the following fractional Schrödinger equation ε 2 s ( − ∆ ) s u + V ( x ) u = P ( x ) f ( u ) + Q ( x ) | u | 2 s ∗ − 2 u in R N , where ε>0 is a parameter, s∈(0 ,1), 2 s ∗ = 2 N N − 2 s , N>2 s, ( − ∆ ) s is the fractional Lapalacian and f is a superlinear and subcritical nonlinearity. Under a local condition imposed on the potential function, combining the penalization method and the concentration-compactness principle, we prove the existence of a positive solution for the above equations.

In this paper, we first study the existence of ground state solutions for the following Schrödinger systems { − ∆ u + V ∞ u = G u ( u , v ) , x ∈ R N , − ∆ v + V ∞ v = G v ( u , v ) , x ∈ R N , u , v > 0 , u , v ∈ H 1 ( R N ) , where N≥3 and G ∈ C 2 ( ( R + ) 2 , R ) . And then, by using variational method and projections on Nehari-Poho z ̆ aev type manifold, we will prove the nonexistence of ground state solutions for the coupled Schrödinger systems { − ∆ u + V ( x ) u = G u ( u , v ) , x ∈ R N , − ∆ v + V ( x ) v = G v ( u , v ) , x ∈ R N , u , v > 0 , u , v ∈ H 1 ( R N ) .