Bound states for semilinear Schrödinger equations with sign-changing potential

Abstract

We study the existence and the number of decaying solutions for the semilinear Schrodinger equations $${-\varepsilon^{2}\Delta u + V(x)u = g(x,u)}$$ , $${\varepsilon > 0}$$ small, and $${-\Delta u + \lambda V(x)u = g(x,u)}$$ , $${\lambda > 0}$$ large. The potential V may change sign and g is either asymptotically linear or superlinear (but subcritical) in u as $${|u| \to \infty}$$ .