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On some Schrödinger equations with non regular potential at infinity

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


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