Multiple Normalized Solutions for Biharmonic Choquard Equation with Hardy-Littlewood-Sobolev Upper Critical and Combined Nonlinearities

In this paper, we study multiple normalized solutions for biharmonic Choquard equation with Hardy-Littlewood-Sobolev upper critical and combined nonlinearities Delta(2)u = lambda u + mu vertical bar u vertical bar(q-2)u + (I-a * vertical bar u vertical bar(4 alpha)*)vertical bar u vertical bar(4 alpha)*(-2)u in R-N, integral(RN) vertical bar u vertical bar(2)dx = a > 0, where N >= 5, mu > 0, 2 < q < 2 + 8/ N, alpha is an element of (0, N), I-alpha is the Riesz potential and 4(alpha)* = N+alpha/N-4. We first show the existence of normalized ground state solutions, and all ground states correspond to local minima of the associated energy functional. Next, when N >= 9, we further assume that 4(alpha)* >= 2, there exist a class of solutions which are not ground states and are located at a mountain-pass level of the energy functional. This study may be the first contribution regarding existence of multiple normalized solutions for biharmonic Choquard equation.