The number of positive solutions for n$n$‐coupled elliptic systems.

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<pij<2∗$2<p_{ij}<2^*$. Moreover, when 2<pij<2∗$2<p_{ij}<2^*$, very different structures of the positive solution set can also be seen in the case where pij$p_{ij}$ close to 2 and the case where pij$p_{ij}$ close to 2∗$2^*$. Similar results are given for elliptic systems with subcritical Sobolev exponents. These results substantially generalize and improve existing results in the literature. To show the effect of the uniqueness result, we apply it to prove existence of a positive solution to a 2‐coupled nonlinear Schrödinger system with critical exponent and LN/2(RN)$L^{N/2}(\mathbb {R}^N)$ potentials. [ABSTRACT FROM AUTHOR]