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On some Schrödinger equations with non regular potential at infinity. (English) Zbl 1193.35061

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.

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
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