## 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.

### MSC:

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