Existence of self-similar solutions to a parabolic system modelling chemotaxis

We investigate a semilinear elliptic equation SE -Deltav - epsilon /2 x . delv = lambdae-1/4\x\(2)e(v) in R-2 with a parameter lambda > 0 and a constant 0 < <epsilon> < 2, and obtain a structure of the pair (<lambda>, v) of a parameter and a solution which dec ays at infinity. This equation arises in the study of self-similar solution s for the Keller-Segel system. Our main results are as follows: (i) There e xists a lambda (*) > 0 such that if 0 < <lambda> < <lambda>(*), (SE) has tw o distinct solutions (v) under bar lambda and (v) over bar lambda satisfyin g (v) under bar lambda < <(v)over bar> lambda, and that if lambda > lambda (*), (SE) has no solution. (ii) If lambda = lambda (*) and 0 < <epsilon> < 1, (SE) has the unique solution v(*); (iii) The solutions <(v)under bar>lam bda and (v) over bar lambda are connected through v(*).