Multiplicity of Normalized Solutions to a Class of Non-autonomous Choquard Equations

In this paper, we consider the multiplicity of solutions to the following Choquard equation -epsilon(2)Delta u+V(x)u=lambda u+epsilon(-alpha)(I-alpha & lowast;[h(x)|u|(p)])h(x)|u|(p-2)u in R-N, with a prescribed mass integral(N)(R)|u|(2)dx=a epsilon(N), where N >= 1, a,epsilon>0, alpha is an element of(0,N), N+alpha/N < p < N+alpha+2/N, I-alpha is the Riesz potential, lambda is an element of R appears as an unknown Lagrange multiplier, h:R-N ->[0,infinity) is a bounded and continuous function and the potential V is a continuous function. Under some assumptions on V, we show that when epsilon is small enough the numbers of normalized ground states are at least the numbers of global maximum points of h.