Stable standing waves of nonlinear Schrodinger equations with potentials and general nonlinearities

The existence and nonexistence of the minimizer of the L-2-constraint minimization problem e(alpha) := inf{E(u)vertical bar u is an element of H1(RN), parallel to u parallel to(2)(L2(RN)) = alpha} are studied. Here, E(u) := 1/2 integral(RN) vertical bar del u vertical bar(2) + V(x)vertical bar u vertical bar(2)dx - integral(RN) F(vertical bar u vertical bar)dx, V(x) is an element of C(R-N), 0 not equivalent to V(x) <= 0, V(x) -> 0 (vertical bar x vertical bar -> infinity) and F(s) = integral(s)(0) f (t)dt is a rather general nonlinearity. We show that there exists alpha(0) >= 0 such that e(a) is attained for alpha > alpha(0) and e(alpha) is not attained for 0 < alpha < alpha(0). We study differences between the cases V(x) not equivalent to 0 and V(x) = 0, and obtain sufficient conditions for alpha(0) = 0. In particular, if N = 1, 2, then alpha(0) = 0, and hence e(alpha) is attained for all alpha > 0.