Cerami, Giovanna; Molle, Riccardo On some Schrödinger equations with non regular potential at infinity. (English) Zbl 1193.35061 Discrete Contin. Dyn. Syst. 28, No. 2, 827-844 (2010). Summary: We study the existence of solutions \(u \in H^1(\mathbb R^N)\) for the problem \(-\Delta u + a(x)u = |u|^{p-2}u\), where \(N \geq 2\) and \(p\) is superlinear and subcritical. The potential \(a \in L^{\infty} (\mathbb R^N)\) is such that \(a(x) \geq c > 0\) but is not assumed to have a limit at infinity. Considering different kinds of assumptions on the geometry of \(a\), we obtain two theorems stating the existence of positive solutions. Furthermore, we prove that there are no nontrivial solutions, when a direction exists along which the potential is increasing. Cited in 7 Documents MSC: 35J61 Semilinear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J10 Schrödinger operator, Schrödinger equation 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B09 Positive solutions to PDEs Keywords:Schrödinger equations; non-regular potential at infinity; variational methods; existence and nonexistence of solutions PDFBibTeX XMLCite \textit{G. Cerami} and \textit{R. Molle}, Discrete Contin. Dyn. Syst. 28, No. 2, 827--844 (2010; Zbl 1193.35061) Full Text: DOI