Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity

In this paper, we are concerned with the existence of solutions to the N-dimensional nonlinear Schrödinger equation −ε²Δu+V(x)u=K(x)up with u(x)>0, u∈H¹(RN), N?3 and . When the potential V(x) decays at infinity faster than ⁻²(1+|x|) and K(x)?0 is permitted to be unbounded, we will show that the positive H¹(RN)-solutions exist if it is assumed that G(x) has local minimum points for small ε>0, here with denotes the ground energy function which is introduced in X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997) 633-655]. In addition, when the potential V(x) decays to zero at most like (1+|x|)^−α with 0<α?2, we also discuss the existence of positive H¹(RN)-solutions for unbounded K(x). Compared with some previous papers A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006) 317-348; A. Ambrosetti, D. Ruiz, Radial solutions concentrating on spheres of NLS with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907; A. Ambrosetti, Z.Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005) 1321-1332] and so on, we remove the restrictions on the potential function V(x) which decays at infinity like (1+|x|)^−α with 0<α?2 as well as the restrictions on the boundedness of K(x)>0. Therefore, we partly answer a question posed in the reference A. Ambrosetti, A. Malchiodi, Concentration phenomena for NLS: Recent results and new perspectives, preprint, 2006].