1. The number of positive solutions for n$n$‐coupled elliptic systems.
- Author
-
Jing, Yongtao, Liu, Haidong, Liu, Yanyan, Liu, Zhaoli, and Wei, Juncheng
- Subjects
POSITIVE systems ,NONLINEAR systems ,NUMBER theory ,MULTIPLICITY (Mathematics) ,EXPONENTS - Abstract
We study the number of positive solutions to the n$n$‐coupled elliptic system −Δui=μiui2∗−1+∑j=1,j≠inβijuipij−1ujqij,ui∈D1,2(RN),i=1,2,...,n,$$\begin{align*} -\Delta u_i&=\mu _iu_i^{2^*-1}+\sum _{j=1,\,j\ne i}^n\beta _{ij}u_i^{p_{ij}-1}u_j^{q_{ij}},\ u_i\in \mathcal {D}^{1,2}(\mathbb {R}^N),\\ &\quad i=1,2,\ldots ,n, \end{align*}$$where N⩾3$N\geqslant 3$, n⩾2$n\geqslant 2$, μi>0$\mu _i>0$, βij>0$\beta _{ij}>0$, pij<2∗$p_{ij}<2^*$, and pij+qij=2∗$p_{ij}+q_{ij}=2^*$ for i≠j∈{1,2,...,n}$i\ne j\in \lbrace 1,2,\ldots ,n\rbrace$. We prove new multiplicity and uniqueness results for positive solutions of the system, whether the system has a variational structure or not. In some cases, we provide a rather complete characterization on the exact number of positive solutions. The results we obtain reveal that the positive solution set of this system has very different structures in the three cases pij<2$p_{ij}<2$, pij=2$p_{ij}=2$, and 2
- Published
- 2024
- Full Text
- View/download PDF