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 Pa:= −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 . On the other hand, if a < a(N), then Pa is not nonnegative in L2(RN) and has a lot of selfadjoint extensions. Here Pa(N) is nonnegative and selfadjoint in L2(RN) in the sense of form-sum. But the energy space D((1 + Pa(N))1/2) does not coincide with H1(RN). 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−itPa(N) which is also proved.
We prove the existence, uniqueness and asymptotic behavior of solutions to the elliptic system −Δu = a(x)f(u,v) in Ω, −Δv = b(x)g(u,v) in Ω, u = v = 0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary ∂Ω,a,b: Ω → (0,∞), and f,g: (0,∞) × (0,∞) → (0,∞) are allowed to be singular at 0 and non-monotone.
We study the global existence of solutions to an n-dimensional parabolic-parabolic system for chemotaxis with a subquadratic degradation. We introduce sublinear production of a chemoattractant. We then show the global existence of solutions in Lp space (p > n) under certain relations between the degradation and production orders.
We consider the Cauchy problem of a parabolic-elliptic system in R2, which appears in various fields in physics and biology. Under a mild restriction on the initial data, we show the global existence of nonnegative solutions to the Cauchy problem for both the subcritical and the critical cases, that is, the total mass of the solution is less than or equal to 8π.
Let Ω be a bounded domain in RN with smooth boundary. Let f: [0, + ∞[ → [0,+∞[, with f(0) = 0, be a continuous function such that, for some a > 0, the function ξ∈]0, +∞[ → ξ−2 · ∫0ξf(t)dt is non increasing in ]0,a[. Finally, let α: Ω → [0,+∞[ be a continuous function with α(x) > 0, for all x ∈ Ω. We establish a necessary and sufficient condition for the existence of solutions to the following problem −Δu = λα(x)f(u) in Ω, u > 0 in Ω, u = 0 on ∂Ω, where λ is a positive parameter. Our result extends to higher dimension a similar characterization very recently established by Ricceri in the one dimensional case.
We prove the unique solvability in the holomorphic category of a system of partial differential equations which is a higher order version of the system considered by Bielawski  in his study of Ricci-flat Kähler metrics. The system involves a higher order nonlinear equation that is singular in the variable t and is very similar to the one studied by Gérard and Tahara  in the 1990s. The proof makes use of a family of majorant functions based on the ones used in Lope-Tahara  and Pongérard .