اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSemitrivial vs. fully nontrivial ground states in cooperative cubic Schrodinger systems with $dge3$ equations
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In this work we consider the weakly coupled Schrodinger cubic system [ begin{cases} displaystyle -Delta u_i+lambda_i u_i= mu_i u_i^{3}+ u_isum_{j eq i}b_{ij} u_j^2 u_iin H^1(mathbb{R}^N;mathbb{R}), quad i=1,ldots, d, end{cases} ] where $1leq Nleq 3$, $lambda_i,mu_i >0$ and $b_{ij}=b_{ji}>0$ for $i eq j$. This system admits semitrivial solutions, that is solutions $mathbf{u}=(u_1,ldots, u_d)$ with null components. We provide optimal qualitative conditions on the parameters $lambda_i,mu_i$ and $b_{ij}$ under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the $d=2$ equations case. For $dgeq 3$ equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case $lambda_iequiv lambda$ and $b_{ij}equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the $d=2$ case.
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