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

In this paper. we are concerned with the existence of solutions to the N-dimensional nonlinear Schrodinger equation -epsilon(2)Delta u + V(x)u = K(x)u(p) with u(x) > 0, u is an element of H(1)(R(N)), N >= 3 and 1 < p < N+2/N-2. When the potential V(x) decays at infinity faster than (1 + |x|)(-2) and K(x) >= 0 is permitted to be unbounded, we will show that the positive H(1)(R(N))-solutions exist if it is assumed that G(x) has local minimum points for small epsilon > 0, here G(x) = V(theta)(x)K(-2/p-1) (x) with theta = p+1/p-1 - N/2 denotes the ground energy function which is introduced in [X. Wang, B. Zeng, On concentration of positive bound states of nonlinear Schrodinger 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|)(-alpha) with 0 < alpha <= 2, we also discuss the existence of positive H(1)(R(N))-solutions for unbounded K(x). Compared with some previous papers [A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of nonlinear Schrodinger 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 Schrodinger 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|)(-alpha) with 0 < alpha <= 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]. (C) 2009 Elsevier Inc. All rights reserved.