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

By Manuel del Pino, M. Kowalczyk, Frank Pacard, Jun-cheng Wei. 2007

69 citations recorded by the source corpus.

We prove the existence of a new class of entire, positive solutions for the classical elliptic problem Δu−u+up=0 in R2, when p>2. The solutions we construct are obtained by perturbing the function ∑j=1kw(dist(⋅,γj)), where k⩾1, w is the unique even, positive, non-constant solution of w″−w+wp=0 in R and where the curves γj are the graphs of the functions f1,…,fk which are solutions of the Toda system c2fj″=efj−1−fj−efj−fj+1 with f0≡−∞ and fk+1≡+∞. This result provides a surprising link between the solutions of the Toda system and entire solutions of the above semilinear elliptic equation.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8bdb8f463e49038dfef2c2b47182df1d73d5921a

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In Applied Mathematics

1. Jun-cheng Wei
2. Yanyan Li
3. Hao Xu
4. Manuel del Pino
5. Jingang Xiong
6. Tianling Jin
7. A. Malchiodi
8. Xu-jia Wang

In Geometry and Topology

1. Hao Xu
2. Jun-cheng Wei
3. Yanyan Li
4. Ludovic Rifford
5. F. Schulze

Citations in the corpus

• Multi-bump solutions of -Δn = K(x)u (n+2)/(n-2) on lattices in ℝ n (2018, methodology): “For compact manifold case, we refer to Del Pino-Kowalczyk-Wei [20, 21] and Pacard-Ritore [45], and for noncompact manifolds, we refer to Del Pino-Kowalczyk-Pacard-Wei [22], Del Pino-Kowalczyk-Wei [19] and the references therein.”