Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations in \(\mathbb {R}^N\) with indefinite weight functions. (English) Zbl 1313.35078

The authors consider the sublinear Schrödinger-Maxwell equation \[ \begin{aligned} &-\Delta u +V(x)u + K(x) \phi u = a(x) | u| ^{q-1}u, \\ &-\Delta \phi = K(x) u^2. \end{aligned} \tag{1} \] Despite the title, both equations are posed in \(\mathbb R^3\). Here, \(q \in (0,1)\) and some hypotheses are given on \(V(x)\), \(K(x)\) and \(a(x)\). The authors use the symmetric mountain pass theorem to prove existence of infinitely many solutions.
The main novelty with respect to other known results is the fact that \(V\) may change sign and does not diverge at infinity, so that compact Sobolev embeddings are not available. This compactness problem is ruled out because \(a(x)\) is negative near infinity under their assumptions.


35J20 Variational methods for second-order elliptic equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)