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June  2010, 28(2): 827-844. doi: 10.3934/dcds.2010.28.827

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, Italy
2. 

Dipartimento di Matematica, Università di Roma "Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy

Received  February 2010 Published  April 2010

In this paper we study the existence of solutions $u\in H^1(\R^N)$ for the problem $-\Delta u+a(x)u=|u|^{p-2}u$, where $N\ge 2$ and $p$ is superlinear and subcritical. The potential $a(x)\in L^\infty(\R^N)$ is such that $a(x)\ge c>0$ but is 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 a direction 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: 35J1.
Citation: Giovanna Cerami, Riccardo Molle. On some Schrödinger equations with non regular potential at infinity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 827-844. doi: 10.3934/dcds.2010.28.827
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