Leray–Lions Equations of (p, q)-Type in the Entire Space with Unbounded Potentials

In this paper we prove the existence of signed bounded solutions for a quasilinear elliptic equation in \({\mathbb {R}}^N\) driven by a Leray–Lions operator of (p, q)–type in presence of unbounded potentials. A direct approach seems to be a hard task, and for this reason we will study approximating problems in bounded domains, whose resolutions needs refined tools from nonlinear analysis. In particular, we will use a weaker version of the classical Cerami–Palais–Smale condition together with a extension of the Weierstrass Theorem due to Candela–Palmieri, as well as a generalization of a celebrated convergence result by Boccardo–Murat–Puel.