On the stationary solutions of the 2D Doi–Onsager model

In this paper, we study the 2D Doi–Onsager model equation(1)F=Z−1e−U(F),on S1 with the potential equation(2)U(F)(ϕ)≔Ul(F)(ϕ)≔β∫S1|sin(θ−ϕ)|lF(θ)dθ, where ll is a positive integer, ββ is a parameter, FF is a probability distribution function.We derive differential equations equivalent to  and  for all positive integers ll, then use the corresponding variational structure to obtain the existence of solutions, in particular, non-constant solutions. When l=1l=1, (1) and (2) are the original Onsager Models, and in this case, we prove that all solutions must be axially symmetric and there exist infinitely many such solutions.