On the Yamabe Equation with Rough Potentials
نویسندگان
چکیده
We study the existence of non–trivial solutions to the Yamabe equation: −∆u+ a(x) = μu|u| 4 n−2 μ > 0, x ∈ Ω ⊂ R with n ≥ 4, u(x) = 0 on ∂Ω under weak regularity assumptions on the potential a(x). More precisely in dimension n ≥ 5 we assume that: (1) a(x) belongs to the Lorentz space L n 2 (Ω) for some 1 ≤ d < ∞, (2) a(x) ≤ M < ∞ a.e. x ∈ Ω, (3) the set {x ∈ Ω|a(x) < 0} has positive measure, (4) there exists c > 0 such that
منابع مشابه
Square-integrability of solutions of the Yamabe equation
We show that solutions of the Yamabe equation on certain ndimensional non-compact Riemannian manifolds which are bounded and Lp for p = 2n/(n−2) are also L2. This Lp-L2-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in our article [1]. As an application we see that the smooth Yamabe invariant of any 2connected compact 7-dimensional ma...
متن کاملGinsburg-Pitaevski-Gross differential equation with the Rosen-Morse and modified Woods-Saxon potentials
In this paper, we consider non-linear Ginsburg-Pitaevski-Gross equation with the Rosen-Morse and modifiedWoods-Saxon potentials which is corresponding to the quantum vortices and has important applications in turbulence theory. We use the Runge- Kutta-Fehlberg approximation method to solve the resulting non-linear equation.
متن کامل2 00 6 Compactness of solutions to the Yamabe problem . III YanYan
For a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates ...
متن کاملCompactness of solutions to the Yamabe problem . III
For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates wou...
متن کاملSecond Yamabe constant on Riemannian products
Let (M, g) be a closed Riemannian manifold (m ≥ 2) of positive scalar curvature and (N, h) any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second N−Yamabe constant of (M × N, g + th) as t goes to +∞. We obtain that limt→+∞ Y (M ×N, [g+ th]) = 2 2 m+n Y (M ×R, [g+ ge]). If n ≥ 2, we show the existence of nodal solutions of the Yamabe equation on (M × ...
متن کامل