Elliptic systems on \(\mathbb R^{N}\) with nonlinearities of linear growth (Q2888967)

This paper discusses existence and multiplicity of solutions for systems \(-\Delta u +V(x) u = \nabla_u F(x,u)\) in \(\mathbb{R}^N\), \(u \in (H^1(\mathbb{R}^N))^m\), with the unknown \(u:\mathbb{R}^N \to \mathbb{R}^m\), \(V\) is an \(m\times m\) matrix valued function, and the potential \(F: \mathbb{R}^{N+m} \to \mathbb{R}\) is close to a quadratic form in \(u\). Under specific conditions on \(V\) and \(F\), the existence of one solution is proved, and if additionally symmetry assumptions are made, then a minimum number of distinct solutions is established.NEWLINENEWLINEFor the entire collection see [Zbl 1237.58002].