Canonical systems of partial differential equations
The author discusses existence results for solutions of some specific elliptic PDE system consisting of two weakly coupled nonlinear Schrödinger equations with saturation effect describing the propogation of light in photonic lattices. It is given by \begin{align*} \Delta v &= \frac{Pv}{1+v^2+w^2} + \lambda v \quad\text{in }\Omega, \\ \Delta w &= \frac{Qw}{1+v^2+w^2} + \lambda w \quad\text{in }\Omega, \end{align*} where \(\lambda,P,Q\in\mathbb{R}\) and \(\Omega\subset\mathbb{R}^n\). In his main results (Theorems~1--6) the author identifies various parameter regimes where the existence of nontrivial solutions can be proved using variational methods such as global minimization (Theorem 1) and well-known Linking Theorems (Theorems~2--6). Reviewer's remark: In the case of ground state solutions (i.e., nontrivial solutions of least energy) more powerful results than the given ones are possible by a comparison of the two scalar least energy levels and the least energy level of the system. If the latter is smaller than the former, the ground state is known to be vectorial, i.e., one has \(v\neq 0\) and \(w\neq 0\). Such arguments have been presented in similar contexts in [Zbl 1286.35104; Zbl 1130.34014; Zbl 1257.35171; Zbl 1104.35053; Zbl 1135.35072; Zbl 1334.35319; Zbl 1312.35082]. Moreover, one may add that vectorial solutions can be obtained by different methods such as bifurcation theory. For the entire collection see [Zbl 1471.34005].
scientific article; zbMATH DE number 7392157
scientific article