Ngô Quốc Anh

April 11, 2012

Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

Filed under: Luận Văn, PDEs, Riemannian geometry — Tags: , — Ngô Quốc Anh @ 2:18

A couple of days ago, I got an acceptance for publication in Advances in Mathematics journal that makes me feel so exciting because of the prestige of the journal. This is part of my PhD thesis in NUS under the supervision of professor Xu. Besides, this is joint work with him.

The work looks like simple, I mean, we just try to solve the following PDE

\displaystyle {\Delta _g}u + hu = f{u^{{2^\star} - 1}} + \frac{a}{{{u^{{2^\star} +1}}}}, \quad u>0,

where \Delta_g=-{\rm div}_g(\nabla_g) is the Laplace-Beltrami operator, 2^\star=\frac{2n}{n-2} is the critical Sobolev exponent, M is a compact manifold without boundary of dimension n \geqslant 3, and h, f, a \geqslant 0 are smooth functions. In our work, the above PDE is numbered as (1.2). I don’t want to mention the physical background of the equation, in a few words, this equation is motivated by the Hamiltonian constraint equations of General Relativity through the so-called conformal method. Apparently, the important and frequently studied prescribing scalar curvature equation is just a particular case.

In this work, we focus on the negative Yamabe-scalar field invariant, that is, h<0. Our result basically consists of two theorems.

In the first result, we consider the case that f may change its sign, we prove

Theorem 1.1. Let (M,g) be a smooth compact Riemannian manifold without the boundary of dimension n \geqslant 3. Assume that f and a \geqslant 0 are smooth functions on M such that \int_M f dv_g<0\sup f > 0, \int_M a dv_g >0, and |h| < \lambda _f where \lambda_f is given in (2.1) below. Let us also suppose that the integral of a satisfies

\displaystyle\int_M {ad{v_g}} < \frac{1}{n-2}{\left( {\frac{{n - 1}}{n-2}} \right)^{n - 1}}{\left( {\frac{{|h|}}{{\int_M {|{f^ - }|d{v_g}} }}} \right)^n}\int_M {|{f^ - }|d{v_g}}

where f^- is the negative part of f. Then there exists a number C > 0 to be specified such that if

\displaystyle\frac{{\sup {f }}}{{\int_M {{|f^ -| }d{v_g}} }} <C,

Equation (1.2) possesses at least two smooth positive solutions.

In the next result, we consider the case that f \leqslant 0. In this case, we are able to get a complete characterization of the existence of solutions. More precisely, we prove

Theorem 1.2. Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n \geqslant 3. Let h<0 be a constant, f and a be smooth functions on M with a \geqslant 0 in M, f \leqslant 0 but not strictly negative. Then Equation (1.2) possesses one positive solution if and only if |h|<\lambda_f.

As one can see, the above theorem does not allow f to be strictly negative. Fortunately, our approach can cover this case too. This is the last remark in the paper as we prove the following: if f<0 then Equation (1.2) always possesses one positive solution, I mean, without any condition on f except the condition f<0.

It is important to note that in the case f \leqslant 0, the solution is always unique by using the monotone trick.

3 Comments »

  1. Thanks for sharing this! Is there a way to obtain a copy of the paper or the thesis?

    Comment by timur — April 12, 2012 @ 1:01

    • Thank you for your interest in my work. Since the paper is just in press, it will be online soon, and I will post a link then once it is available. For the thesis, coz’ I have not submitted it yet so there is no way at the moment. I will put an online version soon after my defense, please look at the list of my publication to find the link.

      Comment by Ngô Quốc Anh — April 12, 2012 @ 1:03

    • The paper is now available online at http://dx.doi.org/10.1016/j.aim.2012.04.007.

      Comment by Ngô Quốc Anh — May 21, 2012 @ 23:02


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