Positive normalized solutions to nonlinear elliptic systems in $\R^4$ with critical Sobolev exponent

Abstract

In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begin{equation}\label{eqA0.1}\nonumber \begin{cases} -Δu+λ_1u=μ_1u^3+α_1|u|^{p-2}u+βv^2u\quad&\hbox{in}~\R^4,\\ -Δv+λ_2v=μ_2v^3+α_2|v|^{p-2}v+βu^2v\quad&\hbox{in}~\R^4,\\ \end{cases} \end{equation} under the mass constraint $$\int_{\R^4}u^2=a_1^2\quad\text{and}\quad\int_{\R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $μ_1,μ_2,β>0$; $α_1,α_2\in \R$, $p\!\in\! (2,4)$ and $λ_1,λ_2\!\in\!\R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $α_i<0(i=1,2)$. Then turning to the case of $α_i>0 (i=1,2)$, if $2<p<3$, we show that the problem admits a ground state and an excited state, which are characterized respectively by a local minimizer and a mountain-pass critical point of the corresponding energy functional. Moreover, we give a precise asymptotic behavior of these two solutions as $(a_1,a_2)\to (0,0)$ and $a_1\sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schrödinger systems with Sobolev critical exponent. When $3\leq p<4$, we prove an existence as well as non-existence ($p=3$) results of the ground states, which are characterized by constrained mountain-pass critical points of the corresponding energy functional. Furthermore, precise asymptotic behaviors of the ground states are obtained when the masses of whose two components vanish and cluster to a upper bound (or infinity), respectively.