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

Abstract

In this paper, we study the nonlocal equation ∫RN|u(x)|2dxγΔu=λu+μ|u|q−2u+|u|p−2u, xinRN having a prescribed mass ∫RN|u(x)|2dx=c2, where N≥3, μ, γ∈(0,+∞), q∈(2,2∗), c is a positive constant, p, q∈(2,2∗) with p≠q and 2∗=2NN−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.