Normalized Solutions for Schrodinger Equations with Stein-Weiss Potential of Critical Exponential Growth

In this paper, we focus on the existence of normalized solutions to the following Schrodinger equation with the Stein-Weiss potential - Delta u +lambda u = ( I-mu x F(u) /| x|(alpha) ) f (u) | x|(alpha), x is an element of R-2, where 2 alpha + mu <= 2, 0 < mu < 2, I-mu denotes the Riesz potential and f : R -> R has critical exponential growth which behaves like eau2. The solutions correspond to critical points of the underlying energy functional subject to the L-2-norm constraint, namely, integral R-2 |u|(2)dx = a(2) for a > 0 given. Under some weak assumptions, we prove the existence of the normalized solution for the equation by developing refined variational methods. In particular, we shall establish two new approaches to estimate precisely the minimax level of the underlying energy functional. As far as we know, our result is the first one in seeking normalized solutions of nonlinear equations involving the nonlocal Stein-Weiss reaction.