Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in \(\mathbb{R}^3\) (Q1947414)

The authors study the nonhomogeneous Schrödinger-Maxwell system \[ -\Delta u+u+\lambda \phi(x)u=|u|^{p-1}u+g(x), \] \[ -\Delta\phi=u^2 \] on \(\mathbb{R}^3\). It is assumed that \(\lambda>0\), \(p\in (1, 5)\) and \(0\leq g(x)\in L^2(\mathbb{R}^3)\) is a radial function. If \(p\in (2, 5)\), then it is shown that for all \(\lambda>0\) there exist two radial solutions with positive and negative energies, respectively, provided the \(L^2\)-norm of \(g\) is bounded above by an explicitly given constant. Similar result holds for \(p\in (1, 2]\) if \(\lambda>0\) is small enough.