Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with non umbilic boundary

1.   Introduction

Let $(M, g)$, a smooth, compact Riemannian manifold of dimension $n\ge7$ with non umbilic boundary. We recall that the boundary of $M$ is called non umbilic if the trace-free second fundamental form of $\partial M$ is everywhere different from zero. Here we study the linearly perturbed problem

Where $\Delta_{g}$ is the Laplace-Beltrami operator and $\nu$ denotes the outer normal. Also, $\varepsilon_{1}, \varepsilon_{2}$ are positive parameters and $\alpha, \beta:M\rightarrow \mathbb{R}$ are smooth functions. We can restate Problem (1.1) in the more compact form

where $L_{g} = \Delta_{g}-\frac{n-2}{4(n-1)}R_{g}$ and $B_{g} = -\frac{\partial}{\partial\nu}-\frac{n-2}{2}h_{g}$.

Problem (1.1) is the perturbed version of the Yamabe boundary problem when the target metric has zero scalar curvature, that is, given a compact Riemannian manifold with boundary, finding a Riemannian metric, conformal to the original one, with zero scalar curvature and constant boundary mean curvature. This represents an extension of the Yamabe problem on manifold with boundary and, since the target metric is conformally flat, also a generalization of the Riemann mapping theorem to higher dimensions. Solving this problem is equivalent to find a positive solution of the equation

which is, as noticed before, the unperturbed version of (1.1). In this paper we study if the perturbation term affects the property of solutions. In particular we want to investigate if the compactness of the set of the solution of the problem holds true for the perturbed problem. Our main results are the following.

Theorem 1. Let $(M, g)$ a smooth, $n$-dimensional Riemannianmanifold of positive type with regular boundary $\partial M$. Supposethat $n\ge7$ and that $\pi(x)$, the trace free second fundamentalform of $\partial M$, is non zero everywhere.

Let $\alpha, \beta:M\rightarrow \mathbb{R}$ smooth functions such that$\beta < 0$ on $\partial M$ and ${ \max_{q\in\partial M}}\{\alpha(q)-\frac{n-6}{4(n-1)(n-2)^{2}}\|\pi(q)\|^{2}\} < 0$.Then, there exist two constants $C > 0$ and $0 < \bar{\varepsilon} < 1$such that, for any $0\le\varepsilon_{1}, \varepsilon_{2}\le\bar{\varepsilon}$and for any $u > 0$ solution of (1.1), it holds

for some $0 < \eta < 1$. The constant $C$ does not depend on $u, \varepsilon_{1}, \varepsilon_{2}$.

Theorem 2. Let $(M, g)$ a smooth, $n$-dimensional Riemannianmanifold of positive type with regular boundary $\partial M$. Supposethat $n\ge7$ and that the trace free second fundamental form of $\partial M$, is non zero everywhere. Let $\alpha, \beta:M\rightarrow \mathbb{R}$ smooth functions.

● If $\beta > 0$ on $\partial M$ then for $\varepsilon_{1}, \varepsilon_{2} > 0$small enough there exists a sequence of solutions $u_{\varepsilon_{1}, \varepsilon_{2}}$of (1.1) which blows up at a suitable point of $\partial M$as $(\varepsilon_{1}, \varepsilon_{2})\rightarrow (0, 0)$.

● If $\beta < 0$ on $\partial M$, $\varepsilon_{1} = 1$, $\alpha > 0$on $M$ and ${ \inf_{q\in\partial M}\alpha(q)+\frac{1}{B}\varphi(q) > 0}$, then for $\varepsilon_{2} > 0$ small enough there exists a sequenceof solutions $u_{\varepsilon_{2}}$ of (1.1) which blowsup at a suitable point of $\partial M$ as $\varepsilon_{2}\rightarrow0$.

Here $B$ and $\varphi(q)$ are defined in Lemma 24.

We remark that in the above Theorem 2, $B$ is strictly positive, $\varphi(q)$ is strictly negative, and both are completely determined by $(M, g)$.

The result of Theorem 1 (and its counterpart Theorem 2) is somewhat unexpected: in classical Yamabe problem ^[1] the compactness of solution is guaranteed as soon as $\alpha$ is negative. In a forthcoming paper we prove that also for boundary Yamabe problem on manifold with umbilic boundary compactness is granted when $\alpha$ is negative while for $\alpha$ positive everywhere there exists a blowing up sequence of solutions. So, this is an example in which the strong analogy between classical Yamabe problem and boundary Yamabe problem breaks down.

The boundary Yamabe problem was firstly introduced by Escobar in ^[2]. Existence results for (1.2) were proved by Escobar ^[2], Marques ^[3], Almaraz ^[4], Brendle and Chen ^[5], Mayer and Ndiaye ^[6]. Solutions of (1.2) could be found by minimization of the quotient

In particular, the solution of is unique, up to symmetries, when $-\infty < Q\le0$ while multiplicity results hold when $Q > 0$. Manifolds for which $Q > 0$ are called of positive type, and it is natural to ask, in that case, when the full set of the solutions of (1.2) forms a $C^{2}$-compact set. This is in complete analogy with classical Yamabe problem. In addition, the celebrated strategy of Khuri, Marques and Schoen ^[7] to prove compactness of solutions of Yamabe problem up to dimension $n = 24$ can be succesfully adapted to Problem (1.2). Indeed, with this method compactness has been proved firstly in the case of locally flat manifolds not conformally equivalent to euclidean balls in ^[8], then for manifolds with non umbilic boundary in ^[9], and, recently, for manifolds with umbilic boundary on which the Weyl tensor does not vanish, in ^[10,11]. These results have been successively extended, but an exhaustive list of reference of compactness results is beyond the scope of this introduction. In ^[12], Almaraz proved that, for $n\ge25$, it is possible to construct umbilic boundary manifolds, not conformally equivalent to euclidean balls, for which Problem (1.2) admits a non compact set of solutions. It is conjectured that also for boundary Yamabe the critical dimension is $n = 24$, but compactness for dimension $n\le24$ is not yet proved in all generality.

As mentioned before, a parallelism arises studying stability of Yamabe problem with respect of small perturbations of curvatures. For classical Yamabe problem, Druet, in the second claim of the main theorem of ^[1], proves that the set of solutions of $-\Delta_{g}u+\frac{n-2}{4(n-1)}a(x)u = cu^{\frac{n+2}{n-2}}\text{ in }M$ is still compact if $a(x)\le R_{g}(x)$ on a manifold $M$ which is not conformally the round sphere and which dimension is $n = 3, 4, 5$. Thus, he claims that in this case the Yamabe problem is stable with respect of perturbation of scalar curvature from below. Also he claims that these results could be extended to higher dimensions. On the other hand, in ^[13,14], Druet, Hebey and Robert found counterexamples to compactness, and so instability, when $a(x)$ is greater than $R_{g}(x)$. In ^[15] the same problem is studied in the case of boundary Yamabe equation by perturbing the mean curvature term, and the same compactness versus blow up phenomenon appeared. So, a first analogy between the role of scalar curvature in classical case and mean curvature in boundary case is established. An analogy between the role of scalar curvature in classical and boundary Yamabe problem when the boundary is umbilic will be investigated by the authors in a forthcoming paper.

As far as we know, Theorem 1 is the first case in which stability is possible when pertrubing a curvature from above, and, therefore, in which the parallelism between classical and boundary Yamabe problem is lost. The result of Theorem 1 is strictly related to non umbilicity of the boundary. In fact, the trace-free second fundamental form competes with the perturbation of the scalar curvature. Thus, when the tensor does not vanish, it could compensate a small positive perturbation. This is clearly observable in Proposition 17, which is a key tool to prove the compactness result (and for the blow-up counterpart, in Lemma 24).

The paper is organized as follows. Hereafter we recall basic definitions and preliminary notions useful to achieve the result. Section 2 is devoted to the proof of the compactness theorem, while in Section 3 we prove the non compactness result.

1.1.   Notations and preliminary definitions

Remark 3 (Notations). We will use the indices $1\le i, j, k, m, p, r, s\le n-1$ and $1\le a, b, c, d\le n$.Moreover we use the Einstein convention on repeated indices. We denoteby $g$ the Riemannian metric, by $R_{abcd}$ the full Riemanniancurvature tensor, by $R_{ab}$ the Ricci tensor and by $R_{g}$ and$h_{g}$ respectively the scalar curvature of $(M, g)$ and the meancurvature of $\partial M$. The bar over an object (e.g., $\bar{R}_{g}$)will means the restriction to this object to the metric of $\partial M$

On the half space $\mathbb{R}_{+}^{n} = \left\{ y = (y_{1}, \dots, y_{n-1}, y_{n})\in\mathbb{R}^{n}, \!\ y_{n}\ge0\right\}$we set $B_{r}(y_{0}) = \left\{ y\in\mathbb{R}^{n}, \!\ |y-y_{0}|\le r\right\}$and $B_{r}^{+}(y_{0}) = B_{r}(y_{0})\cap\left\{ y_{n} > 0\right\}$.When $y_{0} = 0$ we will use simply $B_{r} = B_{r}(y_{0})$ and $B_{r}^{+} = B_{r}^{+}(y_{0})$.On the half ball $B_{r}^{+}$ we set $\partial'B_{r}^{+} = B_{r}^{+}\cap\partial\mathbb{R}_{+}^{n} = B_{r}^{+}\cap\left\{ y_{n} = 0\right\}$and $\partial^{+}B_{r}^{+} = \partial B_{r}^{+}\cap\left\{ y_{n} > 0\right\}$.On $\mathbb{R}_{+}^{n}$ we will use the following decomposition ofcoordinates: $(y_{1}, \dots, y_{n-1}, y_{n}) = (\bar{y}, y_{n}) = (z, t)$where $\bar{y}, z\in\mathbb{R}^{n-1}$ and $y_{n}, t\ge0$.

Fixed a point $q\in\partial M$, we denote by $\psi_{q}:B_{r}^{+}\rightarrow M$the Fermi coordinates centered at $q$. We denote by $B_{g}^{+}(q, r)$the image of $B_{r}^{+}$. When no ambiguity is possible, we willdenote $B_{g}^{+}(q, r)$ simply by $B_{r}^{+}$, omitting the chart$\psi_{q}$.

We recall also that $\omega_{n-2}$ denotes the volume of the $n-1$ dimensional unit sphere $\mathbb{S}^{n-1}$.

At last we introduce here the standard bubble ${ U(y): = \frac{1}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{\frac{n-2}{2}}}}$ which is the unique solution, up to translations and rescaling, of the nonlinear critical problem.

Set

we recall that $j_{1}, \dots, j_{n}$ are a base of the space of the $H^{1}$ solutions of the linearized problem

Given a point $q\in\partial M$, we introduce now the function $\gamma_{q}$ which arises from the second order term of the expansion of the metric $g$ on $M$ (see 1.14). The choice of this function plays a twofold role in this paper. On the one hand, using the function $\gamma_{q}$ we are able to perform the estimates of Lemmas 2.8, 2.9 and Proposition 15. On the other hand, it gives the correct correction to the standard bubble in order to perform finite dimensional reduction.

For the proof of the following Lemma we refer to [9,Prop 5.1] and [16,Proposition 7]

Lemma 4. Assume $n\ge3$. Given a point $q\in\partial M$, thereexists a unique $\gamma_{q}:\mathbb{R}_{+}^{n}\rightarrow \mathbb{R}$a solution of the linear problem

which is $L^{2}(\mathbb{R}_{+}^{n})$-orthogonal to the functions$j_{1}, \dots, j_{n}$ defined in (1.4) and (1.5).

In addition it holds

Finally the map $q\mapsto\gamma_{q}$ is $C^{2}(\partial M)$.

1.2.   Expansion of the metric

It is well known that there exists a metric $\tilde{g}$, conformal to $g$, such that $h_{\tilde{g}}\equiv0$ (see [2,Lemma 3.3]). So, up to a global conformal change of coordinates Problem (1.1) becomes

With this change of coordinates the expansion of the metric is

where $\pi$ is the second fundamental form and $h_{ij}(0)$ are its coefficients, and $\text{Ric}(0) = R_{nini}(0) = R_{nn}(0)$ (see ^[2]).

2.   The compactness result

We start this section by recalling a Pohozaev type identity. This indentity gives us a fundamental sign condition to rule out the possibility of blowing up sequence, as shown in subsection 2.5. A recall of preliminary results on blow up points is collected in subsection 2.2, while a careful analysis of blow up sequences is performed in subsection 2.4. This allows us to conclude the section with the proof of Theorem 1. Throughout this section we work in $\tilde{g}$ metric. For the sake of readability we will omit the tilde symbol in all this section.

2.1.   A Pohozaev type identity

A Pohozaev type identity is often used in Yamabe boundary problem. Here we use the same local version which is introduced in ^[9,10].

Theorem 5 (Pohozaev Identity). Let $u$ a $C^{2}$-solution of the following problem

for $B_{r}^{+} = \psi_{q}^{-1}(B_{g}^{+}(q, r))$ for $q\in\partial M$.Let us define

and

Then $P(u, r) = \hat{P}(u, r)$.

Here $a = 1, \dots, n$, $k = 1, \dots, n-1$ and $y = (\bar{y}, y_{n})$, where$\bar{y}\in\mathbb{R}^{n-1}$ and $y_{n}\ge0$.

2.2.   Isolated and isolated simple blow up points

We collect here the definitions of some type of blow up points, and the basic properties about the behavior of these blow up points (see ^[8,9,19,20]).

Let $\left\{ u_{i}\right\} _{i}$ be a sequence of positive solution to

where $g_{i}\rightarrow g_{0}$ in the $C_{\text{loc}}^{3}$ topology and $0 < \varepsilon_{1, i}, \varepsilon_{2, i} < \bar{\varepsilon}$ for some $0 < \bar{\varepsilon}\le1$. As before, we suppose without loss of generality that $h_{g_{0}}\equiv0$ and $h_{g_{i}}\equiv0$ for all $i$.

Definition 6. 1) We say that $x_{0}\in\partial M$ is a blow up point for the sequence$u_{i}$ of solutions of (2.1) if there is a sequence$x_{i}\in\partial M$ of local maxima of $\left.u_{i}\right|_{\partial M}$such that $x_{i}\rightarrow x_{0}$ and $u_{i}(x_{i})\rightarrow +\infty.$

Shortly we say that $x_{i}\rightarrow x_{0}$ is a blow up point for$\left\{ u_{i}\right\} _{i}$.

2) We say that $x_{i}\rightarrow x_{0}$ is an isolated blow up pointfor $\left\{ u_{i}\right\} _{i}$ if $x_{i}\rightarrow x_{0}$ isa blow up point for $\left\{ u_{i}\right\} _{i}$ and there existtwo constants $\rho, C > 0$ such that

Given $x_{i}\rightarrow x_{0}$ an isolated blow up point for $\left\{ u_{i}\right\} _{i}$, and given $\psi_{i}:B_{\rho}^{+}(0)\rightarrow M$ the Fermi coordinatescentered at $x_{i}$, we define the spherical average of $u_{i}$as

and

for $0 < r < \rho.$

3) We say that $x_{i}\rightarrow x_{0}$ is an isolated simple blowup point for $\left\{ u_{i}\right\} _{i}$ solutions of (2.1)if $x_{i}\rightarrow x_{0}$ is an isolated blow up point for $\left\{ u_{i}\right\} _{i}$and there exists $\rho$ such that $w_{i}$ has exactly one criticalpoint in the interval $(0, \rho)$.

Remark 7. Notice that blow up for elliptic equation with neumann boundary condition often occurs at a point of the boundary (e.g., in the pioneering paper of Ni and Takagi ^[25]). Concerning boundary Yamabe problem, this fact was at first explicitly proved, as in ^[8]. Later on, it was assumed, without loss of generality, that the blow up point $x_0$ as well as the whole sequence $x_i\rightarrow x_0$ belongs to the boundary (see [9,Definition 4.1]),

Given $x_{i}\rightarrow x_{0}$ a blow up point for $\left\{ u_{i}\right\} _{i}$, we set

Obviously $M_{i}\rightarrow +\infty$ and $\delta_{i}\rightarrow0$.

The proofs of the following propositions can be found in ^[4] and in ^[8].

Proposition 8. Let $x_{i}\rightarrow x_{0}$ is an isolated blowup point for $\left\{ u_{i}\right\} _{i}$ and $\rho$ as in Definition 6. We set

Then, given $R_{i}\rightarrow \infty$ and $c_{i}\rightarrow0$, upto subsequences, we have

1. $|v_{i}-U|_{C^{2}\left(B_{R_{i}}^{+}(0)\right)} < c_{i}$;

2. ${ \lim\limits_{i\rightarrow \infty}\frac{R_{i}}{\log M_{i}} = 0}$.

Proposition 9. Let $x_{i}\rightarrow x_{0}$ be an isolatedsimple blow-up point for $\left\{ u_{i}\right\} _{i}$. Let $\eta$small. If $0 < \bar{\varepsilon}\le1$ is small enough and $0 < \varepsilon_{1}, \varepsilon_{2} < \bar{\varepsilon}$, then there exist $C, \rho > 0$ such that

for $y\in B_{\rho}^{+}(0)\setminus\left\{ 0\right\}$ and $k = 0, 1, 2$.Here $\lambda_{i} = \left(\frac{2}{n-2}\right)(n-2-\eta)-1$.

2.3.   A splitting lemma

Here we summarize a result which proves that only a finite number of blow up points may occur to a blowing up sequence of solution. For its proof we refer to [21,Proposition 5.1], [22,Lemma 20], [19,Proposition 1.1], [9,Propositions 4.2 and 8.2].

Proposition 10. Given $K > 0$ and $R > 0$ there exist two constants$C_{0}, C_{1} > 0$ (depending on $K$, $R$ and $(M, g)$) such thatif $u$ is a solution of

and $\max_{\partial M}u > C_{0}$, then there exist $q_{1}, \dots, q_{N}\in\partial M$, with $N = N(u)\ge1$ with the following properties: for $j = 1, \dots, N$

1. set $r_{j}: = Ru(q_{j})^{1-p}$, then $\left\{ B_{r_{j}}\cap\partial M\right\} _{j}$are a disjoint collection;

2. we have $\left|u(q_{j})^{-1}u(\psi_{j}(y))-U(u(q_{j})^{p-1}y)\right|_{C^{2}(B_{2r_{j}}^{+})} < K$(here $\psi_{j}$ are the Fermi coordinates at point $q_{j}$;

3. we have

In addition, if $n\ge7$ and $|\pi(x)|\neq0$ for any $x\in\partial M$, there exists $d = d(K, R)$ such that

Here $\bar{g}$ is the geodesic distance on $\partial M$.

2.4.   Blowup estimates

In this section we provide a fine estimate for the approximation of the rescaled solution near an isolated simple blow up point.

Proposition 11. Let $x_{i}\rightarrow x_{0}$ be an isolated simpleblow-up point for $\left\{ u_{i}\right\} _{i}$ and $\beta < 0$. Then$\varepsilon_{2, i}\rightarrow0$.

Proof. We compute the Pohozaev identity in a ball of radius $r$ and we set $\frac{r}{\delta_{i}} = :R_{i}\rightarrow \infty$.

By Proposition 9 we have that

We estimate now $\hat{P}(u_{i}, r)$. By comparing this term with $P(u_{i}, r)$ we will get the proof.

The terms $I_{3}$ has been estimated in [15,Proposition 8] and it holds

where $B$ is a positive real constant.

For $I_{2}(u_{i}, r)$ we have, by change of variables,

Now, set $I_{m}^{\alpha}: = \int_{0}^{\infty}\frac{s^{\alpha}ds}{\left(1+s^{2}\right)^{m}}$ we have

using the identities $\int_{0}^{\infty}\frac{t^{k}dt}{(1+t)^{m}} = \frac{k!}{(m-1)(m-2)\cdots(m-1-k)}$ and $\int_{0}^{\infty}\frac{dt}{(1+t)^{m}} = \frac{1}{m-1}$. At this point, since $I_{m}^{\alpha} = \frac{2m}{2m-\alpha-1}I_{m+1}^{\alpha}$ and $I_{m}^{\alpha} = \frac{2m-\alpha-3}{\alpha+1}I_{m}^{\alpha+2}$ (see [9,Lemma 9.4]) we have

thus

where $A$ is a real constant.

For the term $I_{1}(u_{i}, r)$ we slightly improve the estimate provided by Almaraz in ^[9]. By the expansion of the metric (1.13), (1.14) and (1.15) we have

By simmetry reasons we have that

since we choose a metric for which the mean curvature of the boundary is zero. So

where $o^{+}(1)$ is a nonnegative constant that vanishes when $i\rightarrow \infty.$

Comparing $\hat{P}(u_{i}, r)$ and $P(u_{i}, r)$, by (2.3), (2.4), (2.5) and (2.6) we get

so

Being $\varepsilon_{1, i} < \bar{\varepsilon} < 1$, the above inequality holds only if $\varepsilon_{2, i}\rightarrow0$.

Since $\varepsilon_{2, i}\rightarrow0$, $\delta_{i}\rightarrow0$ and $\varepsilon_{1, i} < \bar{\varepsilon} < 1$, the proof of the next proposition is analogous to Prop. 4.3 of ^[9].

Proposition 12. Let $x_{i}\rightarrow x_{0}$ be an isolated simpleblow-up point for $\left\{ u_{i}\right\} _{i}$. Then there exist$C, \rho > 0$ such that

1. $M_{i}u_{i}(\psi_{i}(y))\le C|y|^{2-n}$ for all $y\in B_{\rho}^{+}(0)\setminus\left\{ 0\right\}$;

2. $M_{i}u_{i}(\psi_{i}(y))\ge C^{-1}G_{i}(y)$ for all $y\in B_{\rho}^{+}(0)\setminus B_{r_{i}}^{+}(0)$where $r_{i}: = R_{i}M_{i}^{\frac{2}{2-n}}$ and $G_{i}$ is the Green\textquoteright sfunction which solves

and $|y|^{n-2}G_{i}(y)\rightarrow1$ as $|z|\rightarrow0$.

By Proposition 8 and Proposition 12 we have that, if $x_{i}\rightarrow x_{0}$ is an isolated simple blow-up point for $\left\{ u_{i}\right\} _{i}$, then it holds

Which follows is the core of the compacntess claim: we provide the estimates of the blowup profile of an isolated simple blow up point $x_{i}\rightarrow x_{0}$ for a sequence $\left\{ u_{i}\right\} _{i}$ of solutions of (2.1). The strategy to achieve these results is similar to the one contained in [9,Lemma 6.1] and in [15,Section 5], so we will give only the general scheme and emphasize the main differences, while we refer to the cited papers for detailed proofs. Set

we have that $v_{i}$ satisfies

where $\hat{g}_{i}: = g_{i}(\delta_{i}y)$.

Lemma 13. Assume $n\ge7$. Let $\gamma_{x_{i}}$ be definedin (1.7). There exist $R, C > 0$ such that

for $|y|\le R/\delta_{i}$.

Proof. Let $y_{i}$ such that

We can assume, without loss of generality, that $|y_{i}|\le\frac{R}{2\delta_{i}}.$ This will be useful in the next.

By contradiction, suppose that

Defined

we have, by direct computation, that

where

Since $v_{i}\rightarrow U$ in $C_{\text{loc}}^{2}(\mathbb{R}_{+}^{n})$ we have, at once,

We proceed now by estimating $Q_{i}$ and $\bar{Q}_{i}$. As in [9,Lemma 6.1], using the expansion of the metric and the decays properties of $U$ and $\gamma_{x_{i}}$ we obtain

from which we get

In light of (2.9) we also have $A_{i}\in L^{p}(B_{R/\delta_{i}}^{+})$ and $F_{i}\in L^{p}(\partial'B_{R/\delta_{i}}^{+})$ for all $p\ge2$. Since $|w_{i}(y)|\le1$, by (2.9) (2.11), (2.12), (2.14) and by standard elliptic estimates we conclude that $\left\{ w_{i}\right\} _{i}$, up to subesequences, converges in $C_{\text{loc}}^{2}(\mathbb{R}_{+}^{n})$ to some $w$ solution of

Now we prove that $|w(y)|\le C(1+|y|^{-1})$ for $y\in\mathbb{R}_{+}^{n}$. Consider $G_{i}$ the Green function for the conformal Laplacian $L_{\hat{g}_{i}}$ defined on $B_{r/\delta_{i}}^{+}$ with boundary conditions $B_{\hat{g}_{i}}G_{i} = 0$ on $\partial'B_{r/\delta_{i}}^{+}$ and $G_{i} = 0$ on $\partial^{+}B_{r/\delta_{i}}^{+}$. It is well known that $G_{i} = O(|\xi-y|^{2-n})$. By the Green formula and by (2.14) we have

so

Notice that in the third integral we used that $|y|\le\frac{R}{2\delta_{i}}$ to estimate $|\xi-y|\ge|\xi|-|y|\ge\frac{R}{2\delta_{i}}$ on $\partial^{+}B_{R/\delta_{i}}^{+}$. Moreover, since $v_{i}(\xi)\le CU(\xi)$, we get $|w_{i}(\xi)|\le C\frac{\delta_{i}^{n-2}}{\mu_{i}}$ on $\partial^{+}B_{R/\delta_{i}}^{+}$. Hence

For the other terms we use the formula

where $y\in\mathbb{R}^{m+k}\supseteq\mathbb{R}^{m}$, $\eta, l\in\mathbb{N}$, $0 < l < \eta < m$ (see [9,Lemma 9.2] and ^[23,24]), obtaining at last

for $|y|\le\frac{R}{2\delta_{i}}$. By assumption (2.9) we prove

as claimed.

Now we can derive a contradiction. Notice that, since $v_{i}\rightarrow U$ near $0$, and by (1.11) we have $w_{i}(0)\rightarrow0$ as well as $\frac{\partial w_{i}}{\partial y_{j}}(0)\rightarrow0$ for $j = 1, \dots, n-1$. This implies that

It is known (see [9,Lemma 2]) that any solution of (2.15) that decays as (2.18) is a linear combination of $\frac{\partial U}{\partial y_{1}}, \dots, \frac{\partial U}{\partial y_{n-1}}, \frac{n-2}{2}U+y^{b}\frac{\partial U}{\partial y_{b}}$. This, combined with (2.19), implies that $w\equiv0$.

Now, on one hand $|y_{i}|\le\frac{R}{2\delta_{i}}$, so estimate (2.18) holds; on the other hand, since $w_{i}(y_{i}) = 1$ and $w\equiv0$, we get $|y_{i}|\rightarrow \infty$, obtaining

which gives us the contradiction.

Lemma 14. Assume $n\ge7$ and $\beta < 0$. There exists $R, C > 0$such that

for $|y|\le R/\delta_{i}$.

Proof. We proceed by contradiction, supposing that

Thus, by Lemma 13, we have

We define, similarly to Lemma 13,

and we have that $w_{i}$ satisfies (2.10), where $b_{i}$ is as in Lemma 13, and

As before, $b_{i}$ satisfies inequality (2.12) while

so by classic elliptic estimates we can prove that the sequence $w_{i}$ converges in $C_{\text{loc}}^{2}(\mathbb{R}_{+}^{n})$ to some $w$.

We proceed as in Lemma 13 to deduce that, by (2.20)

Now let $j_{n}$ be defined as in (1.5). Similarly to [15,Lemma 12], since $w_{i}$ satisfies (2.10), integrating by parts we obtain

where $\eta_{i}$ is the inward unit normal vector to $\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}$. One can check easily that

Also, since $\beta < 0$, by Proposition 8, we have

and thus

so

By (2.23) and (2.25) we derive a contradiction. Indeed, by the decay of $j_{n}$ and by the decay of $w_{i}$, given by (2.22) and by (2.20), we have

Since $\Delta j_{n} = 0$, one can check that

Also, we can prove that

Finally

since $\frac{\partial j_{n}}{\partial y_{n}}+nU^{\frac{2}{n-2}}j_{n} = 0$ when $y_{n} = 0$.

In light of (2.26) (2.28) and (2.27) we infer, by (2.23), that

Again we have $\alpha(\delta_{i}y)v_{i}(y)\rightarrow \alpha(x_{0})U(y)\text{ for }i\rightarrow +\infty$, so,

Thus

by (2.20). At this point, by (2.25), (2.30) and (2.32), we get

which gives us a contradiction since $B > 0$.

The following proposition is the main result of this section. The proof can be obtained with a first estimate in the spirit of the previous Lemmas, which is iterated until we get the final result. In fact, once one have the result of Lemma 2.9, the proof of the Proposition is very similar to the one of [9,Proposition 6.1]. For the sake of brevity we prefer to omit it.

Proposition 15. Assume $n\ge7$ and $\beta < 0$. Let $\gamma_{x_{i}}$be defined in (1.7). There exist $R, C > 0$ such that

for $|y|\le\frac{R}{2\delta_{i}}$. Here $\tau = 0, 1, 2$ and $\nabla_{\bar{y}}^{\tau}$is the differential operator of order $\tau$ with respect the first$n-1$ variables.

2.5.   Sign estimates of Pohozaev identity terms

We estimate now $P(u_{i}, r)$, where $\left\{ u_{i}\right\} _{i}$ is a family of solutions of (2.1) which has an isolated simple blow up point $x_{i}\rightarrow x_{0}$. This estimate, contained in Proposition 17, is a crucial point when proving the vanishing of the second fundamental form at an isolated simple blow up point.

The leading term of $P(u_{i}, r)$ will be $-\int_{B_{r/\delta_{i}}^{+}}\left(y^{b}\partial_{b}u+\frac{n-2}{2}u\right)\left[(L_{\hat{g}_{i}}-\Delta)v\right]dy$, so we set

and we recall the following result by Almaraz [9,Propositions 5.2 and 7.1].

Lemma 16. For $n\ge7$ we have

where $I_{n}^{n}: = : = \int_{0}^{\infty}\frac{s^{n}ds}{\left(1+s^{2}\right)^{n}}$.

Proposition 17. Let $x_{i}\rightarrow x_{0}$ be an isolated simpleblow-up point for $u_{i}$ solutions of (2.1). Let $\beta < 0$and $n\ge7$. Fixed $r$, we have, for $i$ large

Proof. We remind that the definition of $\hat{P}$ is given in Theorem 1 and we take $v_{i}(y)$ as in (2.7). By Proposition 15 and by (1.8) of Lemma 4, for $|y| < R/\delta_{i}$ we have

so, recalling (2.5) we have

Analogously we obtain

So, for $i$ sufficiently large we obtain

Then, by the estimates on $v_{i}$ obtained in the previous section, using Lemma 2.11, and recalling that, by inequality (1.9), $\int_{\mathbb{R}_{+}^{n}}\gamma_{q}\Delta\gamma_{q}dy\le0$, we get

which gives the proof.

Proposition 18. Assume $n\ge7$, $0\le\varepsilon_{1, i}, \varepsilon_{2, 1}\le\bar{\varepsilon} < 1$, $\beta < 0$ and

Let $x_{i}\rightarrow x_{0}$ be an isolated simple blow-up pointfor $u_{i}$ solutions of (2.1). Then

1. For $i$ large, $\hat{P}(u_{i}, r)\ge\delta_{i}^{2}C_{1}\left[\|\pi(x_{i})\|^{2}\right]+o(\delta_{i}^{2})$for some $C_{1} > 0$;

2. $\|\pi(x_{0})\| = 0.$

Proof. By Proposition 11 and Proposition 9 we have

On the other hand recalling Proposition 17, Theorem 5, the assumption on $\alpha$, and that $\varepsilon_{1, i} < 1$, we have

with $C_{1} > 0$. In addition, we get $\|\pi(x_{i})\|^{2}\le\delta_{i}^{n-4}$, which gives the proof.

Once we have the result of Proposition 17, with strategy similar to 18, we can prove the following Proposition. For a detailed proof we refer to [9,Proposition 8.1].

Proposition 19. Let $x_{i}\rightarrow x_{0}$ be anisolated blow up point for $u_{i}$ solutions of (2.1).Assume $n\ge7$, $0\le\varepsilon_{1, i}, \varepsilon_{2, 1}\le\bar{\varepsilon} < 1$, $\beta < 0$, ${ \max_{q\in\partial M}\left\{ \alpha(q)-\frac{n-6}{4(n-1)(n-2)^{2}}\|\pi(q)\|^{2}\right\} < 0}$and $\|\pi(x_{0})\|\neq0$. Then $x_{0}$ is isolated simple.

2.6.   Proof of Theorem 1

Using what we have obtained throughout this section, we can now prove the compactness result.

Proof of Theorem 1.. By contradiction, suppose that $x_{i}\rightarrow x_{0}$ is a blowup point for $u_{i}$ solutions of (1.1). Let $q_{1}^{i}, \dots q_{N(u_{i})}^{i}$ the sequence of points given by Proposition 10. By Claim 3 of Proposition 10 there exists a sequence of indices $k_{i}\in1, \dots N$ such that $d_{\bar{g}}\left(x_{i}, q_{k_{i}}^{i}\right)\rightarrow0$. Up to relabeling, we say $k_{i} = 1$ for all $i$. Then also $q_{1}^{i}\rightarrow x_{0}$ is a blow up point for $u_{i}$. By Proposition 10 and Proposition 19 we have that $q_{1}^{i}\rightarrow x_{0}$ is an isolated simple blow up point for $u_{i}$. Then by Proposition 18 we deduce that $\|\pi(x_{0})\| = 0$, contradicting the assumption of the theorem. This concludes the proof.

3.   The non compactness result

In this section we perform the Ljapunov-Schmidt finite dimensional reduction, which relies on three steps. First, we start finding a solution of the infinite dimensional problem (3.4) with a ansatz $u = W_{\delta, q}+\delta V_{\delta, q}+\phi$ where $W_{\delta, q}+\delta V_{\delta, q}$ is a model solution and $\phi = \phi_{\delta, q}$ is a small remainder. Then, we study a finite dimensional reduced problem which depends only on $\delta, q$. Finally, we give the proof of Theorem 2.

In the Ljapunov-Schmidt procedure, it will be necessary that $-L_{g}+\varepsilon_{1}\alpha$ is a positive definite operator. Since $-L_{g}$ is positive definite, in the case $\alpha < 0$, we choose $\varepsilon_{1}$ small enough in order to ensure the positivity of $-L_{g}+\varepsilon_{1}\alpha$.

3.1.   The finite dimensional reduction

Since $-L_{g}+\varepsilon_{1}\alpha$ is a positive definite operator, we define an equivalent scalar product on $H^{1}$ as

which leads to the norm $\|\cdot\|_{g}$ equivalent to the usual one.

Given $1\le t\le\frac{2(n-1)}{n-2}$ we have the well known embedding

and we define, by the scalar product $\langle \langle \cdot, \cdot\rangle \rangle _{g}$,

in the following sense: given $f\in L^{\frac{2(n-1)}{n-2}}(\partial M)$ there exists a unique $v\in H^{1}(M)$ such that

At this point Problem (1.1) is equivalent to find $v\in H^{1}(M)$ such that

where

Notice that, if $v\in H_{g}^{1}$, then $f(v)\in L^{\frac{2(n-1)}{n}}(\partial M)$.

Also, problem (1.1) has a variational structure and a positive solution for (1.1) is a critical point for the following functional defined on $H^{1}(M)$

We define a model solution of (1.1) by means of the standard bubble $U$ and of the function $\gamma_{q}$ introduced in Lemma 4

Given $q\in\partial M$ and $\psi_{q}^{\partial}:\mathbb{R}_{+}^{n}\rightarrow M$ the Fermi coordinates in a neighborhood of $q$, we define

where $y = (z, t)$, with $z\in\mathbb{R}^{n-1}$ and $t\ge0$, $\delta x = y = \left(\psi_{q}^{\partial}\right)^{-1}(\xi)$ and $\chi$ is a radial cut off function, with support in ball centered in $0$, of radius $R$. In an analogous way, we define

Finally, given $j_{a}$ defined in (1.4) and (1.5) we define

By means of $\langle \langle \cdot, \cdot\rangle \rangle _{g}$ it is possible to decompose $H^{1}$ in the direct sum of the following two subspaces

and to define the projections

As claimed before, we look for a solution $u_{q}$ of (1.1) having the form

where $\phi\in K_{\delta, q}^{\bot}$. Using $i_{\alpha}^{*}$, (1.1) is equivalent to the following pair of equations

Let us define the linear operator $L:K_{\delta, q}^{\bot}\rightarrow K_{\delta, q}^{\bot}$ as

and a nonlinear term $N(\phi)$ and a remainder term R as

With these operators, the infinite dimensional equation (3.4) becomes

In this subsection we find, for any $\delta, q$ given, a function $\phi$ which solves equation (3.4). Many of the estimates which follow are contained in ^[16], which we refer to for further details. Here we describe only the main steps of each proof.

Lemma 20. Assume $n\ge7$. It holds

Proof. Take the unique $\Gamma$ such that

that is the function solving

We have, by (3.1) that

For $I_{1}$ we have

and direct computation and by the expansions of the metric (1.13) (1.14) we have (see [16,Lemma 9])

Similarly

The proof of estimate for $I_{3}$ is more delicate, and uses in a crucial way that $\gamma_{q}$ solves (1.7). As shown in [16,Lemma 9] we have indeed

which completes the proof.

Lemma 21. Given $(\varepsilon_{1}, \varepsilon_{2})$, for anypair $(\delta, q)$ there exists a positive constant $C = C(\delta, q)$such that for any $\varphi\in K_{\delta, q}^{\bot}$ it holds

This lemma is a standard tool in finite dimensional reduction, so we refer to ^[17,18] for the proof.

Proving that $N$ is a contraction it is also standard. In fact there exists $\eta < 1$ such that, for any $\varphi_{1}, \varphi_{2}\in K_{\delta, q}^{\bot}$ it holds

By Lemma 20, Lemma 21, and by (3.8) we get the last result of this subsection.

Proposition 22. Given $(\varepsilon_{1}, \varepsilon_{2})$, for any pair $(\delta, q)$ there exists a unique $\phi = \phi_{\delta, q}\in K_{\delta, q}^{\bot}$which solves (3.4) such that

In addition the map $q\mapsto\phi$ is $C^{1}$.

Proof. Lemma 21 and (3.8) and by the properties of $i_{\alpha}$, there exists $C > 0$ such that

Now, it is easy to estimate that

By Lemma 20 and by the previous estimates, for the map

it holds

So, it is possible to choose $\rho > 0$ such that $T$ is a contraction from the ball $\|\phi\|_{g}\le\rho(\varepsilon_{2}\delta+\delta^{2})$ in itself. Hence, by the fixed point Theorem, we have the first claim. The second claim is proved by the implicit function Theorem.

3.2.   The reduced functional

Once a solution of Problem (3.4) is found, it is possible to look for a critical point of $J_{g}\left(W_{\delta, q}+\delta V_{\delta, q}+\phi\right)$, solving a finite dimensional problem which depends only on $(\delta, q)$.

Lemma 23. Assume $n\ge7$. It holds

$C^{0}$-uniformly for $q\in\partial M$.

Proof. We have, for some $\theta\in(0, 1)$

All the terms but $\int_{M}\varepsilon_{1}\alpha\left(W_{\delta, q}+\delta V_{\delta, q}\right)\phi d\mu_{g}$ have been estimated in [16,Lemma 12], so we summarize only the key steps. As in Lemma 20, the most delicate term is the nonlinear term on the boundary. In particular we have that

The other terms in (3.10) are easier to estimate and lead to higher order terms.

At last, by Holder inequality we have

so

and we are in position to prove the result.

Lemma 24. Let $n\ge7$. It holds

where

Here $I_{n-1}^{n}: = \int_{0}^{\infty}\frac{s^{n}}{(1+s^{2})^{n-1}}ds$and $\pi(q)$ is the trace free tensor of the second fundamental form.

Proof. The main estimates of this proof are proved in [16,Proposition 13], which we refer for to for a detailed proof. Here, we limit ourselves to estimate the perturbation terms. We have

We easily compute the terms involving $\varepsilon_{1}$ and $\varepsilon_{2}$ taking in account the expansion of the volume form (1.13), getting

and

By direct computation, and by Remark 18 in ^[16] (see also [16,page 1332]) we have that $\int_{\mathbb{R}^{n}}U(y)^{2}dy = \frac{2(n-2)}{(n-1)(n-4)}\omega_{n-2}I_{n-1}^{n}$ and $\int_{\mathbb{R}^{n-1}}U(\bar{y}, 0)^{2}d\bar{y} = \frac{2(n-2)}{n-1}\omega_{n-2}I_{n-1}^{n}$, getting the value of the positive constants $B$ and $C$.

For the remaining terms we refer to [16,Proposition 13] in which is proved that

We conclude by noticing that $\varphi(q)\le0$ by (1.9).

3.3.   Proof of Theorem 2

At first we recall that, in the hypotheses of Theorem 2, we have that the function $\varphi$ defined in Lemma 24 is strictly negative on $\partial M$. Infact $\|\pi(q)\|^{2}$ is non zero by assumption, and $\int_{\mathbb{R}_{+}^{n}}\gamma_{q}\Delta\gamma_{q}$ is non positive by (1.9). With this in mind, we are in position to prove the result.

Proof of Theorem 2.. We start with the first case, $\beta > 0$. We choose

where $\lambda\in\mathbb{R}^{+}$. With this choice, by Lemma 23 and Proposition 22 we have that

and that, by Lemma 24,

We recall a result which is a key tool in Ljapunov-Schmidt procedure, and which is proved, for instance, in [16,Lemma 15]) and which relies on the estimates of Lemma 23.

Remark. Given $(\varepsilon_{1}, \varepsilon_{2}),$ if $(\bar{\lambda}, \bar{q})\in(0, +\infty)\times\partial M$ is a critical point for the reduced functional $I_{\varepsilon_{1}, \varepsilon_{2}}(\lambda, q): = J_{g}\left(W_{\lambda\varepsilon_{2}, q}+\lambda\varepsilon_{2}V_{\lambda\varepsilon_{2}, q}+\phi\right)$, then the function $W_{\bar{\lambda}\varepsilon_{2}, \bar{q}}+\bar{\lambda}\varepsilon_{2}V_{\bar{\lambda}\varepsilon_{2}, \bar{q}}+\phi$ is a solution of (1.1).

To conclude the proof it lasts to find a pair $(\bar{\lambda}, \bar{q})$ which is a critical point for $I_{\varepsilon_{1}, \varepsilon_{2}}(\lambda, q)$.

In this first case we consider $G(\lambda, q): = \lambda\beta(q)C+\lambda^{2}\varphi(q)$. We have that $\beta(q)C$ is strictly positive on $\partial M$, by our assumptions, while, as recalled before, $\varphi$ is strictly negative on $\partial M$. At this point there exists a compact set $[a, b]\subset\mathbb{R}^{+}$ such that the function $G$ admits an absolute maximum in $(a, b)\times\partial M$, which also is the absolute maximum value of $G$ on $\mathbb{R}^{+}\times\partial M$. This maximum is also $C^{0}$-stable, in the sense that, if $(\lambda_{0}, q_{0})$ is the maximum point for $G$, for any function $f\in C^{1}([a, b]\times\partial M)$ with $\|f\|_{C^{0}}$ sufficiently small, then the function $G+f$ on $[a, b]\times\partial M$ admits a maximum point $(\bar{\lambda}, \bar{q})$ close to $(\lambda_{0}, q_{0})$. By the $C_{0}$ stability of this maximum point $(\lambda_{0}, q_{0})$, and by Lemma 24, given $\varepsilon_{2}$ sufficiently small (and $\varepsilon_{1} = o(1)$), there exists a pair $\left(\lambda_{\varepsilon_{1}, \varepsilon_{2}}, q_{\varepsilon_{1}, \varepsilon_{2}}\right)$ which is a maximum point for $I_{\varepsilon_{1}, \varepsilon_{2}}(\lambda, q)$. This implies, in light of the above Remark, that there exists a pair $\left(\bar{\lambda}_{\varepsilon_{1}, \varepsilon_{2}}, \bar{q}_{\varepsilon_{1}, \varepsilon_{2}}\right)$ such that $W_{\bar{\lambda}_{\varepsilon_{1}, \varepsilon_{2}}\varepsilon_{2}, \bar{q}_{\varepsilon_{1}, \varepsilon_{2}}}+\bar{\lambda}_{\varepsilon_{1}, \varepsilon_{2}}\varepsilon_{2}V_{\bar{\lambda}_{\varepsilon_{1}, \varepsilon_{2}}\varepsilon_{2}, \bar{q}_{\varepsilon_{1}, \varepsilon_{2}}}+\phi$ is a solution of (1.1), and the proof for the case $\beta > 0$ is complete.

The proof in the second case is similar. In this case, by assumption, we have that $B\alpha(q)+\varphi(q) > 0$ on $\partial M$. Then we choose

and we obtain that

In this case we define the function $G(\lambda, q)$ as

and, by our assumptions, the coefficient of $\lambda$ is strictly negative on $\partial M$ while the coefficient of $\lambda^{2}$ is strictly positive on $\partial M$, so we can conclude the proof follows in a similar way.

Conflict of interest

The authors declare there is no conflict of interest.