Nonexistence and Existence of Solutions with Prescribed Norms for Nonlocal Elliptic Equations with Combined Nonlinearities

In this paper, we study the nonlocal equation ∫ R N | u ( x ) | 2 d x γ Δ u = λ u + μ | u | q − 2 u + | u | p − 2 u , x in R N having a prescribed mass ∫ R N | u ( x ) | 2 d x = c 2 , where N ≥ 3 , μ , γ ∈ ( 0 , + ∞ ) , q ∈ ( 2 , 2 ∗ ) , c is a positive constant, p , q ∈ ( 2 , 2 ∗ ) with p ≠ q and 2 ∗ = 2 N N − 2 . This research is meaningful from a physical point of view. Using variational methods, we present some results on the nonexistence and existence of solutions under different cases p and q which improve upon the previous ones via topological theory.