The Toda system and multiple-end solutions of autonomous planar elliptic problems

We prove the existence of a new class of entire, positive solutions for the classical elliptic problem Δu−u+up=0Δu−u+up=0 in R2R2, when p>2p>2. The solutions we construct are obtained by perturbing the function∑j=1kw(dist(⋅,γj)), where k⩾1k⩾1, w   is the unique even, positive, non-constant solution of w″−w+wp=0w″−w+wp=0 in RR and where the curves γjγj are the graphs of the functions f1,…,fkf1,…,fk which are solutions of the Toda systemc2fj″=efj−1−fj−efj−fj+1 with f0≡−∞f0≡−∞ and fk+1≡+∞fk+1≡+∞. This result provides a surprising link between the solutions of the Toda system and entire solutions of the above semilinear elliptic equation.