On linearly coupled Schrödinger systems (Q2862193)

The authors consider the following system of nonlinear Schrödinger equations NEWLINE\[NEWLINE\begin{cases} -\Delta u+u=f(u)+\lambda v\( \;\;in \;\)\mathbb{R}^N,\\ -\Delta v+v=g(v)+\lambda u \;\text{ in} \;\mathbb{R}^N,\end{cases}NEWLINE\]NEWLINE \(u(x),v(x)\rightarrow 0\), as \(|x|\rightarrow +\infty\), where \(\lambda> 0\), \(N\geq 3\), and \(f,g\in C(\mathbb{R},\mathbb{R})\). Under the following assumptionsNEWLINENEWLINE-- \(\lim_{s\rightarrow 0}\frac{f(s)}{s}=\lim_{s\rightarrow 0}\frac{g(s)}{s}=0\),NEWLINENEWLINE-- \(\displaystyle{\limsup_{s\rightarrow +\infty}\frac{f(s)}{s^p}<+\infty}\) and \(\displaystyle{\limsup_{s\rightarrow +\infty}\frac{g(s)}{s^p}<+\infty}\), for some \(p\in (1,\frac{N+2}{N-2})\),NEWLINENEWLINE-- \(\displaystyle{\sup_{s>0}\frac{\displaystyle{2\int_0^sf(t)dt}}{s^2}>1}\) and \(\displaystyle{\sup_{s>0}\frac{\displaystyle{2\int_0^sg(t)dt}}{s^2}>1}\),NEWLINENEWLINE the authors prove that there exists \(\lambda_0\in(0,1)\) such that for each \(\lambda\in (0,\lambda_0)\) the above system admits a positive radial solution \((u_\lambda,v_\lambda)\in C^2(\mathbb{R})\times C^2(\mathbb{R})\) with the following property: if \(\{\lambda_n\}\subset (0,\lambda_0)\) is a sequence such that \(\lambda_n\rightarrow 0\), then, up to a subsequence, \((u_{\lambda_n},v_{\lambda_n})\rightarrow (U,V)\) strongly in \(H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)\), where \(U\) and \(V\) are positive radial ground state solutions of the equations \(-\Delta u+u=f(u)\), \(-\Delta v+v=g(v)\), respectively.NEWLINENEWLINEA similar result was proved by the authors in a previous paper where they showed that the sequence \((u_{\lambda_n},v_{\lambda_n})\) strongly converges in \(H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)\) to some \((U,V)\) such that, differently from the present paper, either \(U\equiv 0\) or \(V\equiv 0\). As a consequence, it follows that, for \(\lambda>0\) small enough, the above system must have at least two positive radial solutions.NEWLINENEWLINEThe proof is inspired by a variational method introduced in the paper [\textit{J. Byeon} and \textit{L. Jeanjean}, Arch. Ration. Mech. Anal. 185, No. 2, 185-200 (2007; Zbl 1132.35078)].