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

In this paper, we study the nonlocal equation (integral(RN)vertical bar u(x)vertical bar 2dx)(gamma) Delta u = lambda u + mu vertical bar u vertical bar(q-2)u + vertical bar u vertical bar(p-2)u, x in R-N having a prescribed mass integral(RN) vertical bar u( x)(2)dx = c(2), where N >= 3, mu,gamma is an element of(0,+infinity) q is an element of(2, 2*), c is a positive constant, p, q is an element of (2, 2*) with p not equal q and 2* = 2N/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.