Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: the higher dimensional case

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schrödinger system with $d\ge 2$ equations \begin{equation*} -Δu_{i}+λ_{i}u_{i}=|u_{i}|^{p-2}u_{i}\sum_{j = 1}^{d}β_{ij}|u_{j}|^{p} \text{ in } Ω, \quad u_i=0 \text{ on } \partial Ω, \qquad i=1,...,d, \end{equation*} in the case of a critical exponent $2p=2^{\ast }=\frac{2N}{N-2}$ in high dimensions $N\ge 5$. We treat the focusing case (${\beta }_{ii}>0$ for every $i$) in the variational setting ${\beta }_{ij}={\beta }_{ji}$ for every $i\ne j$, dealing with a Brézis-Nirenberg type problem: $-{\lambda }_1(\Omega )<{\lambda }_i<0$, where ${\lambda }_1(\Omega )$ is the first eigenvalue of $(-\Delta ,H_0^1(\Omega ))$. We provide several sufficient conditions on the coefficients ${\beta }_{ij}$ that ensure the existence of least energy positive solutions; these include the situations of pure cooperation (${\beta }_{ij}>0$ for every $i\ne j$), pure competition (${\beta }_{ij}\le 0$ for every $i\ne j$) and coexistence of both cooperation and competition coefficients. Some proofs depend heavily on the fact that $1<p<2$, revealing some different phenomena comparing to the special case $N=4$. Our results provide a rather complete picture in the particular situation where the components are divided in two groups. Besides, based on the results about a phase separation phenomena, we prove the existence of least energy sign-changing solution to the Brézis-Nirenberg problem \[ -Δu+λu=μ|u|^{2^*-2}u,\quad u\in H^1_0(Ω), \] for $\mu >0$, $-{\lambda }_1(\Omega )<\lambda <0$ for all $N\ge 4$, a result which is new in dimensions $N=4,5$.

32 pages