Multiple solutions for Schrödinger-Maxwell systems with unbounded and decaying radial potentials (Q2877637)

In this paper the authors study the following system of PDEs. NEWLINE\[NEWLINE-\Delta u+V(|x|)u+Q(|x|)\phi u=Q(|x|)f(u)\text{, }x\in\mathbb{R}^{3}NEWLINE\]NEWLINE NEWLINE\[NEWLINE-\Delta\phi=Q(|x|)u^2\text{, }x\in\mathbb{R}^{3}NEWLINE\]NEWLINE The authors demonstrate existence of solution to this problem under a variety of conditions on the maps \(Q\), \(V\in\mathcal{C}((0,+\infty);[0,+\infty))\) and \(f\). The conditions are too numerous to mention properly here, but it is worth mentioning that \(V\) and \(Q\), for example, are required to satisfy conditions of the sort NEWLINE\[NEWLINE\liminf_{r\to0}\frac{V(r)}{r^{a_0}}>0\text{ and }\liminf_{r\to+\infty}\frac{V(r)}{r^{a_1}}>0,NEWLINE\]NEWLINE for constants \(a_0\) and \(a_1\), and NEWLINE\[NEWLINE\limsup_{r\to0}\frac{Q(r)}{r^{b_0}}>0\text{ and }\limsup_{r\to+\infty}\frac{Q(r)}{r^{b_1}}>0,NEWLINE\]NEWLINE for constants \(b_0\) and \(b_1\). The techniques utilized by the authors are variational in nature.