Existence of Solutions to the Logarithmic Choquard Equations in High Dimensions

In this paper, we consider the following logarithmic Choquard equation -Delta u + a(x)u + lambda(log vertical bar.vertical bar * vertical bar u vertical bar(2))u = b vertical bar u vertical bar(p-2)u, in R-n, where n >= 3, lambda > 0, 2 < p < 2n/n-2, a is an element of L-infinity (R-n). For n = 2, p > 2, this equation has been studied extensively. In this paper, we prove the existence of a mountain-pass solution and a ground state solution for n >= 3 by using the variational method.