On coupled systems of Schrödinger equations
The system of linearly coupled semilinear Schrödinger equations \[ \begin{cases} -\Delta u+u &= f(x,u)+\lambda v, \qquad x\in \mathbb{R}^{N},\\ -\Delta v+v &= g(x,v)+\lambda u, \qquad x\in \mathbb{R}^{N},\end{cases} \] is studied. The following two cases are considered: (i) \(f(x,u)=(1+a(x)) |u|^{p-1}u\), \(g(x,v)=(1+a(x)) |v|^{p-1}v\), \(1<p<2^{\star} -1\), where \(2^{\star}\) is the critical Sobolev exponent, and (ii) \(f(x,u)=f(u)\), \(g(x,u)=g(u)\). The authors prove, under some assumptions on \(f\) and \(g\), the existence of a positive bound state in case (i) or of a positive ground state in case (ii) for \(\lambda \in (0,1)\). They also analyse the limit behavior of the positive bound or ground state as \(\lambda \rightarrow 0\) and obtain energy estimates of the state as \(\lambda\) changes. The existence of a positive ground state was previously proved in case (i) in [\textit{A. Ambrosetti, G. Cerami} and \textit{D. Ruiz}, J. Funct. Anal. 254, No. 11, 2816--2845 (2008; Zbl 1148.35080)], and in case (ii) in [\textit{H. Brézis} and \textit{E. H. Lieb}, Commun. Math. Phys. 96, 97--113 (1984; Zbl 0579.35025)]. Nevertheless, the methods of proof used in the paper under review allow a more thorough study of the dependence on \(\lambda\) of the bound or ground states and of their energy.
coupled system of semi-linear Schrödinger equations
scientific article; zbMATH DE number 5953773