Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth

In this paper, we study the following fractional Schrödinger equation with critical or supercritical growth( − Δ ) s u + V ( x ) u = f ( x , u ) + λ | u | p − 2 u , x ∈ R N , where 0  <   s   <  1, N   >  2 s , λ   >  0,2 s ∗ = 2 N N − 2 s , p ≥ 2 s ∗ , ( − Δ) s denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical function. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ   >  0 by variational methods. Our main contribution is related to the fact that we are able to deal with the case p > 2 s ∗ .