Normalized solutions for the Choquard equations with critical nonlinearities

This study is concerned with the existence of normalized solutions for the Choquard equations with critical nonlinearities {-Delta u + lambda u = f(u) + (I-a *; |u|(2)(a)*)|u| (2a*-2), in R-N, integral(N)(R) |u|(2)dx = a(2), where N > 2 , alpha is an element of (0, N), a > 0 , and I-alpha (x) is the Riesz potential given by I-alpha (X) = A(a)/|X|(N-a) with A(a) = Gamma[N - alpha/ 2]/2(alpha)pi(N/2) Gamma[alpha/2], and = 2(a)* = N + alpha/N - 2 is the Hardy-Littlewood-Sobolev critical exponent and f is a subcritical nonlinearity. In the case - that f is L-2-supercritical growth, by means of the Pohozaev manifold method and mountain pass theorem, we obtain a couple of the normalized solution; while in the case f(u) = mu|u|(q-2) u with 2 < q < 2 + 4/N, being L 2-subcritical growth and mu > 0 a parameter, we employ the truncation technique and the genus theory to prove the multiplicity of normalized solutions.