On some Schrodinger equations with non regular potential at infinity

In this paper we study the existence of solutions u is an element of H(1) (R(N)) for the problem -Delta u + a(x)u = vertical bar u vertical bar(p-2)u, where N >= 2 and p is superlinear and subcritical. The potential a(x) is an element of L(infinity) (R(N)) is such that a(x) >= 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.