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].