Normalized Solutions to Schrodinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities

We study the existence and nonexistence of normalized solutions (u(a), lambda(a)) is an element of H-1(R-N) x R to the nonlinear Schrodinger equation with mixed nonlocal nonlinearities: integral - Delta u = lambda u + (I-alpha * vertical bar u vertical bar(p))vertical bar u vertical bar(p-2) u + mu(I-alpha * vertical bar u vertical bar(q))vertical bar u vertical bar(q-2) u in R-N, integral(RN) |vertical bar u vertical bar(2)dx = a(2) > 0, where N >= 3, alpha is an element of(0, N), mu is an element of R, N+alpha/N < q < p <= N+alpha/N-2, and I-alpha is the Riesz potential. This study can be viewed as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions to nonlocal Schrodiger equations with a fixed L-2-norm parallel to u parallel to(2) = a > 0. The leading term is L-2-supercritical, that is, p is an element of(N+alpha+2/N, N+alpha/N-2], where the Hardy-Littlewood-Sobolev critical exponent p = N+alpha/N-2 appears. We first prove that there exist two normalized solutions if q is an element of (N+alpha/N, N+alpha+2/N) with mu > 0 small, that is, one is at the negative energy level while the other one is at the positive energy level. For q = N+a+2 N, we show that there is a normalized ground state for 0 < mu < (mu) over tilde and there exist no ground states for mu > (mu) over tilde, where (mu) over tilde is a sharp positive constant. If q is an element of(N+alpha+2/N, N+alpha/N-2), we deduce that there exists a normalized ground state for any mu > 0. We also obtain some existence and nonexistence results for the case mu < 0 and q is an element of( N+alpha/N, N+alpha+2/N]. Besides, we analyze the asymptotic behavior of normalized ground states as mu -> 0(+).