Solvability of Nonlinear Schrödinger Equations with Some Critical Singular Potential via Generalized Hardy-Rellich Inequalities

Abstract

Nonlinear Schrödinger equations with inverse-square potentials (NLS)_a are considered. Since the potential |x|⁻² is quite singular, the scaling argument does not work well. In view of the selfadjointness of P_a:= −Δ + a|x|⁻², a = a(N):= −(N−2)²/4 seems to be the threshold of the unique solvability. In fact, if a > a(N), then the unique solvability for (NLS)_a is proved by the energy methods established by Okazawa-Suzuki-Yokota [12]. On the other hand, if a < a(N), then P_a is not nonnegative in L²(R^N) and has a lot of selfadjoint extensions. Here P_a(N) is nonnegative and selfadjoint in L²(R^N) in the sense of form-sum. But the energy space D((1 + P_a(N))^1/2) does not coincide with H¹(R^N). Thus we identify the energy space by applying generalized Hardy-Rellich inequalities. By virtue of the identification we can apply the energy methods and conclude the global solvability for (NLS)_a with a = a(N), the critical coefficient. Moreover, the uniqueness can be shown by using the Strichartz estimates for e^−itP_a(N) which is also proved.