On some Schrödinger equations with       non regular potential at infinity

Giovanna Cerami; Riccardo Molle

doi:10.3934/dcds.2010.28.827

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2010, Volume 28, Issue 2: 827-844. Doi: 10.3934/dcds.2010.28.827 open access

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On some Schrödinger equations with non regular potential at infinity

Giovanna Cerami^1, and
Riccardo Molle^2,

1.
Dipartimento di Matematica, Politecnico di Bari, Via E. Orabona, 4, 70125 Bari
2.
Dipartimento di Matematica, Università di Roma "Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma

Received: January 31, 2010

Published: June 2010

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Abstract

In this paper we study the existence of solutions $u\in H^1(\R^N)$ forthe problem $-\Delta u+a(x)u=|u|^{p-2}u$, where $N\ge 2$ and $p$ issuperlinear and subcritical.The potential $a(x)\in L^\infty(\R^N)$ is such that $a(x)\ge c>0$ butis not assumed to have a limit at infinity.Considering different kinds of assumptions on the geometry of $a(x)$we obtain two theorems stating the existence of positive solutions.Furthermore, we prove that there are no nontrivial solutions, when adirection exists along which the potential is increasing.

Keywords:

Schrödinger equations,
non-regular potential at infinity,
variational methods,
existence and nonexistence of solutions.

Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35J10.

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