Existence of solitary waves for supercritical Schrödinger systems in dimension two (Q745565)

The authors are concerned with the existence of variational solutions for the system \(-\Delta u+a(x)u=g(v)\) in \(\mathbb R^2\), \(-\Delta v+a(x) v=f(u)\) in \(\mathbb R^2\), where \(a\) is a continuous function that is bounded away from zero such that \(a^{-1/q}\in L^{2,p}(\mathbb R^2)\), \(a^{-1/p}\in L^2(\mathbb R^2)\). It is assumed that \(f,g\in C[0,\infty)\) are nonnegative, \(f(t), g(t)=o(t)\) near the origin and there exists \(\theta>2\) such that \(\theta F(t)\leq tf(t)\) and \(\theta G(t)\leq tg(t)\) for all \(t>0\). Under some extra conditions on \(f\) and \(g\), the authors obtain the existence of a nontrivial solution both in the critical and subcritical case with respect to the Trudinger-Moser growth.