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تسجيل مستخدم جديدPositive normalized solutions to nonlinear elliptic systems in $R^4$ with critical Sobolev exponent
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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} onumber begin{cases} -Delta u+lambda_1u=mu_1u^3+alpha_1|u|^{p-2}u+beta v^2uquad&hbox{in}~R^4, -Delta v+lambda_2v=mu_2v^3+alpha_2|v|^{p-2}v+beta u^2vquad&hbox{in}~R^4, end{cases} end{equation} under the mass constraint $$int_{R^4}u^2=a_1^2quadtext{and}quadint_{R^4}v^2=a_2^2,$$ where $a_1,a_2$ are prescribed, $mu_1,mu_2,beta>0$; $alpha_1,alpha_2in R$, $p!in! (2,4)$ and $lambda_1,lambda_2!in!R$ appear as Lagrange multipliers. Firstly, we establish a non-existence result for the repulsive interaction case, i.e., $alpha_i<0(i=1,2)$. Then turning to the case of $alpha_i>0 (i=1,2)$, 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 $a_1sim a_2$. This seems to be the first contribution regarding the multiplicity as well as the synchronized mass collapse behavior of the normalized solutions to Schr{o}dinger systems with Sobolev critical exponent. When $3leq p<4$, 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.
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