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Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. (English) Zbl 1480.35241

Let \((M^n,g)\) be an \(n\)-dimensional compact Riemannian manifold of positive type not conformally equivalent to the standard ball with regular umbilic boundary \(\partial M^n\), \(R_g\) be the scalar curvature of \(M^n\), \(L_g=\Delta_g-\frac{n-2}{4(n-1)}R_g\), \(h_g\) be the mean curvature of the \(\partial M^n\), and \(\nu\) be the outer normal unit vector to \(\partial M^n\). The authors examine the properties of a solution of the following boundary value problem: \[\begin{cases}&L_gu=0\quad\text{in }M^n,\\&\frac{\partial u}{\partial\nu}+\frac{n-2}{2}h_gu+\varepsilon\alpha u=(n-2)u^{n/(n-2)}\quad\text{on }\partial M^n,\end{cases}\tag{1}\]where \(\alpha\) is a real smooth function on \(M^n\) and negative on \(\partial M^n\) and \(\varepsilon>0\). To be precise, they state that for a given \(\bar{\varepsilon}>0\), there is a positive constant \(C\) such that for \(\varepsilon\in(0,\bar\varepsilon)\) and for a positive solution \(u\) of \((1)\) it holds \(\frac{1}{C}\le u\le C\) and \(\|u\|_{C^{2,\eta}(M^n)}\le C\) for some \(\eta\in (0,1)\). The constant \(C\) does not rely neither on \(u\) nor on \(\varepsilon\) and whenever the Weyl tensor does not vanish on \(\partial M^n\) and \(n\ge 8\). Also, they state an analogous result when the manifold is not endowed with umbilic boundary and for \(n\ge 7\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
58J05 Elliptic equations on manifolds, general theory
35B20 Perturbations in context of PDEs
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References:

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