The first GJMS invariant

Abstract

Let (M, g) be a smooth compact Riemannian manifold of dimension $n\ge 3$. Denote by $P_{g}$ the GJMS (Graham–Jenne–Mason–Sparling) operator. In this paper, we introduce the GJMS invariant $\mu$ and we define the first GJMS invariant $\mu _1$ as the infimum of the first eigenvalue of $P_{g}$ over the metrics conformal to g and of volume 1. We study when it is attained and whether is equal to $\mu$ . As an application, we show that the nonlinear GJMS equation $P_{g}v =\mu _{1}|v|^{N-2}v$ has nodal (sign-changing) solution. When g is Einstein, the above equation has positive solutions if the scalar curvature $S_{g}>0$.

1 Introduction

Let (M, g) be a compact Riemannian manifold of dimension $n\ge 3$, and let k be an integer such that $k \ge 1$ and $2k \le n$. In 1992, in [15] Graham–Jenne–Mason–Sparling have defined a family of conformally invariant differential operators defined for any Riemannian metric (GJMS operators for short). The construction of these operators is based on the ambient metric of Fefferman–Graham [13]. More precisely, for any Riemannian metric g on M, there exists a local, formally self-adjoint, conformally covariant operator

$$\begin{aligned} P_g : C^\infty (M) \longrightarrow C^\infty (M), \end{aligned}$$

such that for all $u\in$ $C^{\infty }(M)$, the GJMS operator $P_{g}$ is given by :

$$\begin{aligned} P_{g}u=\Delta _{g}^{k}u+lot \end{aligned}$$

(1)

where $\Delta _{g}$ is the Laplace-Beltrami operator, and lot denotes differential terms of lower order. For more detail about $P_g$, we refer the reader to Robert [24]. This operator enjoys nice conformal invariance properties. Indeed, let $\varphi \in C^{\infty }(M)$ be a positive function and $N=\frac{2n}{n-2k}$. If $n \ne 2k$, then any metric $\overline{g}$ written in the form $\varphi ^{\frac{4}{n-2k}}g$ is a conformal metric to g and therefore, for any metric $\overline{g}$ conformal to g, the operator $P_g$ is conformally invariant in the following sense: for all $u\in C^{\infty }(M)$, we have $P_{g}(u\varphi )=\varphi ^{N-1}P_{\overline{g}}(u)$. By taking $u\equiv 1$, we get

$$\begin{aligned} P_{g}\varphi =\frac{n-2k}{2}Q_{\overline{g}}\varphi ^{N-1} \end{aligned}$$

(2)

where

$$\begin{aligned} Q_{\overline{g}}=\frac{2}{n-2k}P_{\overline{g}}(1). \end{aligned}$$

The scalar $Q_g$ is called the Q-curvature and is a Riemannian invariant associated to this operator. Historically, the notion of the Q-curvature is due to Branson’s 1995 article in Transactions of the AMS see [7]. He also defined it in the critical case $n = 2k$. Now when $k=1$, $P_{g}$ is the conformal Laplacian operator and $Q_g$ is the scalar curvature $S_g$ (up to a constant). The problem of prescribing a constant scalar curvature is known as the Yamabe problem, the classical reference for this problem is a survey by Lee-Parker [19]. When $k=2$, $P_{g}$ is the Paneitz-Branson operator introduced by Paneitz in [22] and the Q-curvature was introduced by Branson-Ørsted [8]. Results for the prescription of the Q-curvature problem for the Paneitz operator are in Djadli-Hebey-Ledoux [10], Robert [23], Esposito-Robert [12], Hang-Yang [17], Gursky-Malchiodi [16] and Benalili-Boughazi [3]. Moreover, concerning fourth-order problems, there has been also an intensive literature on the question, we refer the reader to [3, 5, 9, 23]. Solving the problem of prescribing Q-curvature for the GJMS operator is a very difficult problem and its underlying analysis is intricate, we refer to Robert [24] and Mazumdar [21] for some particular situations. The simple case of these problems is prescribing constant Q-curvature which is equivalent to finding a positive smooth solution u of the following equation

$$\begin{aligned} P_{g}u = C \vert u \vert ^{N-2}u \end{aligned}$$

(3)

where C is a constant. In order to obtain solutions, we define the quantity

$$\begin{aligned} \mu =\underset{u \in C^{\infty }(M), u > 0}{\inf }I(u) \end{aligned}$$

(4)

where

$$\begin{aligned} I(u)=\frac{\int \nolimits _{M}uP_{g}udv_{g}}{(\int \nolimits _{M}|u |^{N}dv_{g})^{\frac{2}{N}}}. \end{aligned}$$

(5)

As in the Yamabe problem, the constant $\mu$ will be called the GJMS invariant. In particular, if $u \in C^{\infty }(M)$, $u > 0$ and satisfy $I(u) = \mu$, clearly u is solution of (3) and $\overline{g}=u^{\frac{4}{n-2k}}g$ is the desired metric of constant Q-curvature. It is well known that the operator $P_{g}$ is elliptic, self-adjoint with respect to the inner product in $L^2(M)$ [24] and has discrete spectrum with eigenvalues

$$\begin{aligned} \lambda _{1}(g) \le \lambda _{2}(g) \le \lambda _{3}(g)\cdots \le \lambda _{k}(g) \rightarrow +\infty \end{aligned}$$

appear with their multiplicities. The variational characterization of the first eigenvalue $\lambda _{1}(g)$ of $P_{g}$ is given by:

$$\begin{aligned} \lambda _{1}({g})=\underset{v \in H_{k}^{2}(M),u\ne 0}{ \inf }\frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}v^{2}dv_{g}}. \end{aligned}$$

(6)

where the space $H_{k}^{2}(M)$ is the completion of $C^{\infty }(M)$ for the norm

$$\begin{aligned} \Vert u \Vert _{H_{k}^2} = \left( \int _{M}\sum _{l=0}^{k} \vert \nabla ^{l}u \vert ^{2}dv_{g}\right) ^\frac{1}{2}. \end{aligned}$$

(7)

Now by referring to Ammann-Humbert [1], we introduce an invariant $\mu _{1}$ that we will call the first GJMS invariant and we will define it by:

$$\begin{aligned} \mu _{1}=\underset{\overline{g}\in [g]}{\inf }\lambda _{1}(\overline{g}) vol(M,\overline{g})^{^{\frac{2k}{n}}} \end{aligned}$$

(8)

where the set $[g] = \lbrace \overline{g} = u^{\frac{4}{n-2k}}g, u \in C^{\infty }(M) \text {and} u > 0 \rbrace$ is the conformal class of the metric g and $vol(M,\overline{g})= \int _M u^Ndv_g$ denotes the Riemannian volume of M with respect to the metric $\overline{g}$.

In order to find minimizers, we enlarge the conformal class [g] to what we call the class of generalized metrics conformal to g. We say that $\overline{g}=u^{\frac{4}{n-2k}}g$ is a generalized metric of the Riemannian metric g if $u \in L^{N}(M), u \ge 0$ and u is not identically null. By the standard minimax method via Rayleigh quotients for defining eigenvalues combined with conformal covariance of $P_{g}$, one sees that for any generalized metric $\overline{g}=u^{\frac{4}{n-2k}}g$, the first eigenvalue $\lambda _{1}(\overline{g})$ of the GJMS opreator $P_{g}$ is characterized by

$$\begin{aligned} \lambda _{1}(\overline{g})=\underset{V\in Gr_{1}^{u}(H_{k}^{2}(M))}{\inf \text { }}\underset{\text { }v\in V\backslash \{0\}}{\text { }\sup } \frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}u^{N-2}v^{2}dv_{g}} \end{aligned}$$

(9)

where the Grassmannian $Gr_{1}^{u}(H_{k}^{2}(M))$ is given in the Definition (2.2).

The purpose of this paper is to study the first eigenvalue $\lambda _{1}(\overline{g})$ for any generalized metric $\overline{g}$ and the main problem is whether the first GJMS invariant $\mu _{1}$ is attained by a generalized metric (or conformal metric) and is equal to the GJMS invariant $\mu$. To solve this problem, we will use the ideas from [1,2,3,4,5,6, 11, 18]. More precisely, the method we would like to apply is introduced in [1] for studying the second Yamabe invariant $\mu _2$ (see Definition (2.1) for $\mu _2$) and generalized for the Paneitz-Branson operator on Einstein manifolds by Benalili and Boughazi in [3]. For clarity purposes, we state our main generic theorem and after we give some results about this method in the next section:

Theorem 1.1

Let (M, g) be a compact Riemannian manifold of dimension $n \ge 3$. Assume that $\lambda _{1}(g) > 0$ and $\mu < {K_{0}}^{-1}$ where $\mu$ is the GJMS invariant and $K_{0}$ is the best constant in the Sobolev embedding of $H_{k}^{2}(M)$. Then there exists a nontrivial function $v \in C^{2k}(M)$ which satisfies $P_{g}v=\mu _{1} \vert v \vert ^{N-2}v$. In other words, $\mu _{1}$ is attained by the generalized metric $\overline{g}=\vert v \vert ^{\frac{4}{n-2k}}g$ and in particular, if $Q_{g} \le 0$, v is a nodal (sign-changing) solution. Moreover, if g is Einstein and $S_g > 0$, the solution $v > 0$ and $v \in C^{\infty }(M)$ and this implies that $\mu = \mu _{1}$ and means that $\overline{g}$ is a conformal metric. Consequently, in the latter case $\mu _{1}$ is attained by the desired metric $\overline{g}$ of constant Q-curvature: $Q_{\overline{g}}=\frac{2}{n-2k}\mu _1$.

Note that this theorem is a consequence of Theorem (3.1), Proposition (3.2) and Theorems (4.1), (5.1) and (6.1). The remainder of this paper is organized as follows: In Sect. 2, we give a short motivation by recalling some results and we quote some facts which we will use in the sequel. In Sect. 3, we establish some results concerning the eigenvalues; in particular, if $\lambda _{1}(g)> 0$, for all generalized metric $\overline{g} = u ^{\frac{4}{n-2k}}g$, the first eigenvalue $\lambda _{1}(\overline{g})$ is achieved, we also show that the linear equation $P_{g}v =\lambda _{1}(\overline{g})u^{N-2}v$ has nodal (sign-changing) solution if $Q_{g} \le 0$ and if $\lambda _{1}(g) < 0$, we show that there exists a generalized metric $\overline{g}$ such that $\lambda _{1}(\overline{g})=-\infty$ which implies that $\mu _1=-\infty$. In Sect. 4, we study the first GJMS invariant $\mu _{1}$ in case $\lambda _{1}(g) > 0$, we will prove that $\mu _{1}$ is attained by a generalized metric if $1- \mu K_{0}>0$ where $\mu$ is the GJMS invariant and $K_{0}$ is the best constant in the Sobolev embedding see (20). In Sect. 5, we show that the nonlinear GJMS equation $P_{g}v =\mu _{1}\vert v \vert ^{N-2} v$ has a nodal solution if $Q_{g} \le 0$. In Sect. 6, we deal with Einstein manifold. In particular, when $S_g > 0$, we will prove that the solution v of the latter equation is positive, $\mu _{1}=\mu$ and is attained by a conformal metric $\overline{g}$ which leads to a metric with constant Q-curvature and in the case $S_{g} < 0$, the solution v is nodal. At the end, we show that $K_{0}=(\mu (S^n,h))^{-1}$ where $\mu (S^n,h)$ is the GJMS invariant of the standard unit n-sphere of $\mathbb {R}^{n+1}$ and there are certain manifolds such that the assumption $1- \mu K_{0}>0$ holds.

Note that when (M, g) is Einstein manifold with positive scalar curvature, the Q-curvature $Q_{g}=\frac{n-2k}{2}(\prod ^{k}_{l=1} c_l) (S_g )^k$, is constant and positive [the reals $c_l$ are given in (17)]. Therefore it is easy to see that $u = 1$ is solution of $P_{g}u=\mu _{1}u^{N-1}$ if $\mu _{1}=\frac{n-2k}{2}Q_{g}$, in other word $\mu _1$ is achieved by the metric g. If $\mu _{1}\ne \frac{n-2k}{2}Q_{g}$ and $1-\mu K_{0}>0$ Theorem (6.1) proves the existence of $u \in C^{\infty }(M), u>0$ solution to the latter equation. In particular, it follows the existence of a metric $\overline{g} = u^{\frac{4}{n-2k}}g$ which is different from the initial g and such that $Q_{\overline{g}}$ is constant. But it is not clear whether the solution u is different from the trivial constant solution. This question seems to be hard. However, when $P_g$ is the conformal Laplacian operator the reader is refereed to [18] for more detail on the question.

2 Motivation and generality

We start by giving a short motivation by recalling some results. Indeed, in [1] Ammann and Humbert defined the Yamabe invariant of high order $\mu _{p}$ by

Definition 2.1

$$\begin{aligned} \mu _{p}=\underset{\overline{g}\in [g]}{\inf }\lambda _{p}(\overline{g}) \left[ vol(M,\overline{g})\right] ^{\frac{2}{n}} \end{aligned}$$

where

$$\begin{aligned} \lambda _{p}(\overline{g})=\underset{V\in Gr_{p}^{u}(H_{k}^{2}(M))}{\inf \text { }}\underset{\text { }v\in V\backslash \{0\}}{\text { }\sup } \frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}u^{N-2}v^{2}dv_{g}} \end{aligned}$$

is the $p^{th}$ eigenvalue of the conformal Laplacian $P_{g}$, $\overline{g}=u^{\frac{4}{n-2k}}g$ is a generalized metric, $p\in \mathbb {N}^{*}$ and the Grassmannian $Gr_{p}^{u}(H_{k}^{2}(M))$ is given in the Definition (2.2).

The authors studied the second Yamabe invariant $\mu _{2}$ in the case $\mu \ge 0$ where $\mu$ is the Yamabe invariant. In particular, they obtained the following theorem:

Theorem 2.1

Let (M, g) be a compact Riemannian manifold of dimension $n \ge 3$. Assume that $\mu _{2}$ is attained by a generalized metric. Then the following equation $P_{g}w=\mu _{2}u^{N-2}w$ has a nodal solution $w \in C^2(M)$ such that $u = \vert w \vert$.

Inspired by the previous results. In [3], Benalili and Boughazi generalize this method to the Paneitz-Branson operator on Einsteinian manifolds. Under some assumptions, they studied $\mu$, $\mu _{1}$ and $\mu _{2}$ in the case $S_{g}>0$ and after ten years the authors in [5] extend these results to the case $S_{g} <0$. For more detail and similar work, we refer the readers to Benalili-Boughazi [4], Boughazi [6] and S. Elsayed [11]. We also specify a very interesting result proven in [11] which states that the sign of eigenvalue $\lambda _{p}(\overline{g})$ is conformal invariant. Clearly, in this paper we try to find similar results with the GJMS operator. More precisely, we study $\mu _1$ and we show in which case we can get $\mu _1=\mu$. Note that the study of $\mu _2$ seems to be much more difficult.

In the following, we quote some facts which will be used in this paper. Put $L_{+}^{N}(M) = \{u\in L^{N}(M), u \ge 0 \text {and} u\ne 0\}$ and $C_{+}^{\infty }(M) =\lbrace u\in C^{\infty }(M), u > 0\rbrace$.

Definition 2.2

For all $u\in L_{+}^{N}(M)$, $p\in \mathbb {N}^{*}$, the Grassmannian $Gr_{p}^{u}(H_{k}^{2}(M))$ is the set of all subspaces of $H_{k}^{2}(M)$ of dimension p and such that the restriction operator to $M \backslash u^{-1}(0)$ is injective. More explicitly, we have the subspace $V=span(v_{1},\ldots ,v_{p})\in Gr_{p}^{u}(H_{k}^{2}(M))$ if and only if the functions $\ v_{1},\ldots ,v_{p}$ are linearly independent on $M \backslash u^{-1}(0)$. Sometimes it will be convenient to use the equivalent statement that the functions $u^{\frac{4}{n-2k}}v_1, u^{\frac{4}{n-2k}}v_2,\ldots ,u^{\frac{4}{n-2k}}v_p$ are linearly independent.

Remark 2.1

The number $N =\frac{2n}{n-2k}$ is known as the critical exponent of the Sobolev embedding which [18], asserts that the space $H_{k}^{2}(M)\subset L^{q}(M)$ where $1< q\le N$ and this embedding is compact when $q < N$.

Definition 2.3

A generalized metric conformal to g is a metric of the form $\overline{g}=u^{\frac{4}{n-2k}}g$ such that $u\in L_{+}^{N}(M)$ where the space $L_{+}^{N}(M)$ is defined in the bottom of the previous page.

Now, we give some properties of the GJMS operator. For the proofs, the reader is refereed to Robert [24] and the references therein. The operator $P_g$ can be written (partially) in divergence form, we precise this divergence form that will be useful in the sequel:

Proposition 2.1

Let $P_g$ be the conformal GJMS operator. Then for any $l \in \{ 1,\ldots ,k-1\}$, there exists $A_{(l)}(g)$ a smooth $T_{2l}^0$-tensor field on M such that

$$\begin{aligned} P_g v= \Delta _{g}^{k}v+\sum \limits _{l=1}^{k-1} A_{l,g} (v)+ \frac{n-2k}{2}Q_{g}v \end{aligned}$$

(10)

where

$$\begin{aligned} A_{l,g} (v) = (-1)^l \nabla ^{j_l...j_1} ( A_{(l)}(g)_{i_1...i_lj_1...j_l} \nabla ^{i_1...i_l}v). \end{aligned}$$

(11)

Indices are raised via the musical isomorphism. In addition for any $l \in \{ 1,\ldots ,k-1\}$, $A_{(l)}(g)$ is symmetric in the following sense: $A_{(l)}(g)(X,Y)=A_{(l)}(g)(Y,X)$ for all $T_{0}^l$-tensors X, Y on M. In particular, for all $u, v \in C^{\infty }(M)$ we have

$$\begin{aligned} \int \nolimits _{M} vP_{g}udv_g = \int \nolimits _{M} uP_{g}vdv_g = \int \nolimits _{M} \Delta _{g}^{\frac{k}{2}}u \Delta _{g}^{\frac{k}{2}}v + \sum \limits _{l=0}^{k-1}A_{(l)}(g) (\nabla ^{l}u, \nabla ^{l}v)dv_{g} \end{aligned}$$

(12)

where for $l=0$, $A_{(0)}(g) (\nabla ^{0}u, \nabla ^{0}v)=\frac{n-2k}{2}Q_{g}uv$. Here, we have adopted the convention

$$\begin{aligned} \Delta _{g} ^{\frac{k}{2}}(u) = \left\{ \begin{array}{l} \Delta _{g}^{m}(u) \text {if} \quad k=2m \quad \text {is even}\\ \nabla \Delta _{g}^{m}(u) \quad \text {if} \quad k=2m+1 \quad \text {is odd} \end{array} \right. \end{aligned}$$

and, when $k=2m+1$ is odd, $\Delta _{g}^{\frac{k}{2}}u \Delta _{g}^{\frac{k}{2}}v = (\nabla \Delta _{g}^{m}u,\nabla \Delta _{g}^{m}v)$.

Since $A_{(l)}(g)$ are smooth, then for any $l \in \{ 0,\ldots ,k-1\}$, there exist $C_l>0$ such that for all $u \in H_{k}^{2}(M)$, one has

$$\begin{aligned} |\int _M \sum \limits _{l=0}^{k-1}A_{(l)}(g) (\nabla ^{l}u, \nabla ^{l}u)dv_{g} |\le \sum \limits _{l=0}^{k-1}C_l \int _M |\nabla ^{l}u |^2 dv_{g} \le \max (C_l) \Vert u \Vert _{H_{k-1}^2}^2 \end{aligned}$$

(13)

As a consequence of (12), we get that the bilinear form $(u, v) \mapsto \int _{M} uP_g v dv_g$ extends to a continuous symmetrical bilinear form on the space $H_{k}^{2}(M) \times H_{k}^{2}(M)$.

We say that $P_g$ is coercive if there exists $C > 0$ such that

$$\begin{aligned} \int \nolimits _{M} v P_g v dv_{g}\ge & {} C \Vert v\Vert _{2}^2 \quad \forall v \in H_{k}^{2}(M). \end{aligned}$$

(14)

Proposition 2.2

For all $u \in H_{k}^{2}(M)$, we define the semi-norm $\Vert u \Vert _{P_g}$ by

$$\begin{aligned} \Vert u \Vert _{P_g}=\left( \int _{M}uP_{g}udv_{g}\right) ^{\frac{1}{2}}. \end{aligned}$$

(15)

Assume that $P_{g}$ is coercive. Then $\Vert . \Vert _{P_g}$ is a norm on $H_{k}^{2}(M)$ equivalent to the standard norm $\Vert . \Vert _{H_{k}^2}$. In addition, if $(v_m)_m$ is a sequence in ${H_{k}^{2}}(M)$ such that $v_m \longrightarrow 0$ weakly in ${H_{k}^{2}}(M)$, and $v_m \longrightarrow 0$ strongly in ${H_{k-1}^{2}}(M)$, then from Bochner-Lichnerowicz-Weitzenbock type formula, one gets that

$$\begin{aligned} \int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v_m)\vert ^{2}dv_{g} = \int \nolimits _{M} \vert \nabla ^{k}v_m\vert ^{2}dv_{g}+o(1), \end{aligned}$$

(16)

The next definition can be found in [18].

Definition 2.4

A Riemannian manifold (M, g) is Einstein, if and only if there exists a real $\lambda$ such that the Ricci tensor writes $Ric_{g}=\lambda g$. Here $\lambda = \frac{S_{g}}{n}$, where $S_{g}$ is the scalar curvature and is constant in this case.

The reader is refereed to [14] for the two following propositions:

Proposition 2.3

Assume that (M, g) is Einstein, then $P_g$ expresses as an explicit product of second-order operators with constant coefficients that depend only on the scalar curvature. In other words, the GJMS operator $P_g$ is given by

$$\begin{aligned} P_g =\prod ^{k}_{l=1} (\Delta _{g} +c_l S_g ) \quad \text {where} \quad c_l=\frac{(n+2l-2)(n-2l)}{4n(n-1)}. \end{aligned}$$

(17)

Moreover, by calculating one can write

$$\begin{aligned} P_g= \Delta _{g}^{k}+\sum \limits _{l=0}^{k-2}b_{k-l-1}(S_g)^{l+1} \Delta _g^{k-l-1}+ b_0 (S_g)^{k} \end{aligned}$$

(18)

where $b_{k-1},\ldots , b_1, b_0$ are positive real numbers obtained from $c_l$.

In addition, formula (12) implies that

$$\begin{aligned} \int _{M}uP_{g}udv_{g} =\int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(u) \vert ^{2} + \sum \limits _{l=0}^{k-1}b_{k-l-1}(S_g)^{l+1} \vert \nabla ^{k-l-1}u \vert ^{2})dv_{g}. \end{aligned}$$

(19)

Proposition 2.4

Assume that the metric g is Einstein with $S_{g} > 0$ and $n > 2k$, then $P_{g}$ is coercive and for all $u\in C^{2k}(M)$ such that $P_{g}u \ge 0$, either $\ u > 0$ or $\ u\equiv 0$.

In this definition, we are going to introduce the best constant in the Sobolev embedding $D_{k}^{2}(\mathbb {R}^{n})\subset L^{N}(\mathbb {R}^{n})$. The reader is refereed to Lions [20].

Definition 2.5

Let $D_{k}^{2}(\mathbb {R}^{n})$ be the space defined as the completion of $C_{c}^{\infty }(\mathbb {R}^{n})$ for the norm $\Vert \Delta ^{\frac{k}{2}}u \Vert _{2}$. It is well know that

$$\begin{aligned} {K_{0}}^{-1}=\underset{u\in D_{k}^{2}\mathbb {R}^{n})-\left\{ 0\right\} }{\inf }\frac{\int _{\mathbb {R}^{n}} \vert \Delta ^{\frac{k}{2}}u \vert ^{2}dv_{g}}{(\int _{\mathbb {R} ^{n}}|u |^{N}dv_{g})^{\frac{2}{N}}} \end{aligned}$$

(20)

and $K_{0}$ is the best constant in the Sobolev’s continuous embedding $D_{k}^{2}(\mathbb {R}^{n}) \subset L^{N}(\mathbb {R}^{n})$. It follows from Sobolev’s embedding theorem that $K_{0}>0$. Moreover, the infimum is achieved by $U : x \longmapsto (1 + \vert x\vert ^2 )^{k- \frac{n}{2}}$, and that all minimizers are compositions of U by translations, homotheties and dilatations.

We also introduce the following results. For the proofs, the reader is refereed to Mazumdar [21].

Theorem 2.2

Let (M, g) be a compact Riemannian manifold of dimension n and let k be a positive integer such that $2k < n$. For any $\epsilon > 0$, there exists $B_{\epsilon } > 0$ such that for all $u \in H_{k}^{2}(M)$ one has

$$\begin{aligned} \Vert u \Vert _{N}^{2} \le (K_{0} + \epsilon ) \int \nolimits _{M} \vert \Delta _{g} ^{\frac{k}{2}}(u)\vert ^2dv_g + B_{\epsilon } \Vert u \Vert _{H_{k-1}^{2}}^2. \end{aligned}$$

(21)

where $K_{0}$ is given by formula (20).

Moreover, for all $v \in C^{\infty }(M)$, there exists $C > 0$ (depend on $\Vert v \Vert _{\infty }$) such that

$$\begin{aligned} \int \nolimits _{M} \vert \Delta _{g}^{\frac{k}{2}}(vu)\vert ^2dv_g \le C \int \nolimits _{M} \vert \Delta _{g} ^{\frac{k}{2}}(u)\vert ^2dv_g + \Vert u \Vert _{H_{k-1}^2}^2 \end{aligned}$$

(22)

where $u \in H_{k}^2(M)$.

Proposition 2.5

Let (M, g) be a closed manifold of dimension n and let k be a positive integer such that $2k < n$. Let $f \in C^{0,\alpha }(M)$ a H$\ddot{o}$lder continuous function. Suppose that $u \in H_{k}^2(M)$ be a weak solution of $P_g u = f|u |^{N-2} u$. Then $u \in C^{2k}(M)$, and is a classical solution of the above equation. Further if $u > 0$ and $f \in C^{\infty }(M)$, then $u \in C^{\infty }(M)$.

3 Generalized metrics and the first eigenvalue

Theorem 3.1

For any generalized metric $\overline{g} = u^{\frac{4}{n-2k}}g$, assume that $u > 0$. Then any normalized minimizing sequence of $\lambda _{1}(\overline{g})$ is bounded in $H_{k}^{2}(M)$.

Proof

Let $(v_m)_m$ be a minimizing sequence of $\lambda _{1}(\overline{g})$, in other words

$$\begin{aligned} \lambda _{1}(\overline{g})=\lim _{m \longrightarrow +\infty } \lambda _{1,m} \quad \text {where} \quad \lambda _{1,m}=\frac{\int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v_m)\vert ^{2}+\sum \limits _{l=0}^{k-1} A_{(l)}(g) (\nabla ^{l}v_m, \nabla ^{l}v_m))dv_{g}}{\int \nolimits _{M}u^{N-2}v_m^{2}dv_{g}}. \end{aligned}$$

It is easy to see that $(\lambda v_{m})_{m}$ is also a minimizing sequence, then if we choose $\lambda = (\int \nolimits _{M}u^{N-2} v_m^{2}dv_{g})^{-\frac{1}{2}}$, it follows that $\int \nolimits _{M}u^{N-2}(\lambda v_m)^{2}dv_{g}=\lambda ^{2}\int \nolimits _{M}u^{N-2} v_m^{2}dv_{g}=1$, hence the sequence $(\lambda v_{m})_{m}$ is renormalized. Without loss of generality, we assume that the sequence $(v_m)_m$ is such that

$$\begin{aligned} \int \nolimits _{M}u^{N-2}v_m^{2}dv_{g}=1. \end{aligned}$$

(23)

1) If $\lambda _{1}(\overline{g})> 0$, then for all v in $H_{k}^{2}(M) \smallsetminus \lbrace 0 \rbrace$, one has

$$\begin{aligned} \int \nolimits _{M} v P_g v dv_{g}\ge & {} \lambda _{1}(\overline{g})\int \nolimits _{M}u^{N-2}v^{2}dv_{g}\\\ge & {} \underbrace{\lambda _{1}(\overline{g}) \underset{x \in M}{\min } u(x)^{N-2}}_{C} \int \nolimits _{M}v^{2}dv_{g}\quad \text {since} \quad u > 0\\\ge & {} C \Vert v \Vert _{2}^{2} \end{aligned}$$

this means that $P_g$ is coercive. Then Proposition (2.2) implies that $\Vert . \Vert _{P_g}$ is a norm on $H_{k}^{2}(M)$ equivalent to the standard norm $\Vert . \Vert _{H_{k}^2}$, then for m large enough, one has

$$\begin{aligned} \lambda _{1,m}=\int \nolimits _{M} v_m P_g v_m dv_{g}= \Vert v_{m}\Vert _{P_g}^{2} \le \lambda _{1}(\overline{g})+1 , \end{aligned}$$

hence the sequence $(v_{m})_m$ is bounded in $H_{k}^{2}(M)$ and $\lambda _{1,m} \ge 0$.

2) If $\lambda _{1}(\overline{g})< 0$, the GJMS operator is not necessarily coercive, then we will assume that $(v_m)_m$ is not bounded in $H_{k}^{2}(M)$, in other words $\Vert v_m \Vert _{H_{k}^2} \longrightarrow +\infty$ and we let

$$\begin{aligned} {v'_m}=\frac{v_m}{\Vert v_m \Vert _{H_{k}^2}}. \end{aligned}$$

Clearly ${\Vert v'_m \Vert }_{H_{k}^2} = 1$, this means that the sequence $(v'_m)_m$ is bounded in $H_{k}^{2}(M)$ and after restriction to a subsequences still labeled $(v'_m)_m$, we may assume that there exists $v' \in H_{k}^{2}(M)$ such that $v'_m \longrightarrow v'$ weakly in ${H_{k}^{2}}(M)$ and $v'_m \longrightarrow v'$ strongly in $H_{k-1}^{2}(M)$.

On the other hand, the sequence $(v'_m)_m$ satisfies the following equation:

$$\begin{aligned} \int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v'_m)\vert ^{2}dv_{g}+\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v'_m, \nabla ^{l}v'_m)dv_{g} =\lambda _{1,m} \int \nolimits _{M}u^{N-2}{v'}_m^{2}dv_{g}. \end{aligned}$$

(24)

Now from the weak convergence, we have

$$\begin{aligned} \lim \int \nolimits _{M}u^{N-2}v'v'_mdv_{g} =\int \nolimits _{M}u^{N-2}(v')^{2}dv_{g}, \end{aligned}$$

and since

$$\begin{aligned} 0 \le \int \nolimits _{M}u^{N-2}(v'-v'_m)^{2}dv_{g} = \int \nolimits _{M}u^{N-2}(v')^{2}dv_{g} -2 \int \nolimits _{M}u^{N-2}v'v'_mdv_{g} +\int \nolimits _{M}u^{N-2}(v'_m)^{2}dv_{g} \end{aligned}$$

one has,

$$\begin{aligned} \int \nolimits _{M}u^{N-2}(v')^{2}dv_{g} \le \int \nolimits _{M}u^{N-2}(v'_m)^{2}dv_{g} = \frac{\int _{M}u^{N-2}v_{m}^{2}dv_{g}}{\Vert v_m \Vert _{H_{k}^2}^2} = \frac{1}{\Vert v_m \Vert _{H_{k}^2}^2}\longrightarrow 0. \end{aligned}$$

(25)

Consequently,

$$\begin{aligned} \int \nolimits _{M}u^{N-2}(v')^{2}dv_{g} = 0 \end{aligned}$$

and since $u>0$, it is easy to see that

$$\begin{aligned} v'\equiv 0. \end{aligned}$$

It follows that $v'_m \longrightarrow 0$ weakly in ${H_{k}^{2}}(M)$ and $v'_m \longrightarrow 0$ strongly in $H_{k-1}^{2}(M)$ therefore,

$$\begin{aligned} \int \nolimits _{M}\sum \limits _{l=0}^{k-1} \vert \nabla ^{l}v'_m \vert ^{2}dv_{g} \longrightarrow 0 \end{aligned}$$

(26)

then by (26) and using (13), one has also

$$\begin{aligned} \sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v'_m, \nabla ^{l}v'_m)dv_{g} \longrightarrow 0. \end{aligned}$$

(27)

Again by (26), the following equality

$$\begin{aligned} 1= \Vert v'_m \Vert _{H_{k}^2}^2 = \int \nolimits _{M} \vert \nabla ^{k}v'_m \vert ^{2}dv_{g} + \int \nolimits _{M}\sum \limits _{l=0}^{k-1} \vert \nabla ^{l}v'_m \vert ^{2}dv_{g} \end{aligned}$$

leads necessarily to

$$\begin{aligned} \int \nolimits _{M} \vert \nabla ^{k}v'_m \vert ^{2}dv_{g} \longrightarrow 1. \end{aligned}$$

Independently, from formula (16) i.e

$$\begin{aligned} \int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v'_m)\vert ^{2} dv_{g}= \int \nolimits _{M} \vert \nabla ^{k}v'_m\vert ^{2}dv_{g}+o(1), \end{aligned}$$

this implies that

$$\begin{aligned} \int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v'_m)\vert ^{2}dv_{g} \longrightarrow 1. \end{aligned}$$

(28)

Since $\lambda _{1}(\overline{g})< 0$, then for m large enough $\lambda _{1,m} < 0$, it follows from (24), (25), (27) and (28) that the sequence $(v'_m)_m$ is such that

$$\begin{aligned} \underbrace{\int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v'_m)\vert ^{2}dv_{g}}_{\longrightarrow 1} + \underbrace{\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v'_m, \nabla ^{l}v'_m)dv_{g}}_{\longrightarrow 0} = \underbrace{\lambda _{1,m}\int \nolimits _{M}u^{N-2}(v'_m)^{2}dv_{g}}_{ \longrightarrow a }, \end{aligned}$$

where $a \le 0$ or does not exist, in all cases this gives a contradiction. This proves that $(v_m)_m$ is bounded in $H_{k}^{2}(M)$.

Moreover, we have

$$\begin{aligned} -\int \nolimits _{M} \vert \nabla ^{k}v_m \vert ^{2}dv_{g} \le \int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v_m)\vert ^{2}dv_{g}, \end{aligned}$$

which lead to

$$\begin{aligned}&-\int \nolimits _{M} \vert \nabla ^{k}v_m \vert ^{2}dv_{g} - \max (C_l) \Vert u \Vert _{H_{k-1}^2}^2 \le \int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v_m)\vert ^{2}dv_{g}\\&\quad + \sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v_m, \nabla ^{l}v_m)dv_{g} \end{aligned}$$

where $\max (C_l)$ is given by (13), this means that

$$\begin{aligned} \min (-1,- \max (C_l)) \Vert v_m \Vert _{H_{k}^2}^2 \le \int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v_m)\vert ^{2}+\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v_m, \nabla ^{l}v_m)dv_{g} \end{aligned}$$

In other words,

$$\begin{aligned} \lambda _{1,m} \ge \min (-1,-\max (C_l)) \Vert v_m \Vert _{H_{k}^2}^2 \end{aligned}$$

and since $(v_m)_m$ is bounded, then there exists $M > 0$ such that

$$\begin{aligned} \lambda _{1,m} \ge \min (-1,-\max (C_l)) M >-\infty . \end{aligned}$$

$\square$

Proposition 3.1

Assume that $\lambda _{1}(g)<0$, then there exists $u\in L_{+}^{N}(M)$ such that $\lambda _{1}(\overline{g})=-\infty$ where $\overline{g}=u^{\frac{4}{n-2k}}g$.

Proof

Since $\lambda _{1}(g)<0$, there exist a function $v\in C^{\infty }(M)$ such that $\int \nolimits _{M}vP_gvdv_{g}<0$. Fix a point p in M. For $\epsilon >0$, let $\phi _ {\epsilon }$ be a cut-off function adapted to our context, in other words a smooth function such that :

$$\begin{aligned} \left\{ \begin{array}{l} 0 \le \phi _{\epsilon }\le 1 \\ \phi _{\epsilon }=0 \quad \text {on} \quad B_{\epsilon }(p) \quad \text {and} \quad \phi _{\epsilon }=1 \quad \text {on} \quad M\backslash B_{2\epsilon }(p)\\ \vert \nabla ^l\phi _{\epsilon }\vert \le \frac{c_l}{\epsilon ^l } \quad \text {for all} \quad l \quad \text {in} \quad \lbrace 1,2, ...,k-1 \rbrace (28_1) \\ \vert \Delta _{g}^ {\frac{k}{2}} \phi _{\epsilon } \vert \le \frac{c_k}{\epsilon ^k } (28_2) \end{array} \right. \end{aligned}$$

where $B_{\epsilon } (p)$ is the open ball centered at p and of radius $\epsilon$ and $c_l>0$ are constants that do not depend on $\epsilon$.

We claim that:

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \int _M (\phi _{\epsilon }v)P_{g}(\phi _{\epsilon }v)dv_g = \int _M v P_{g} vdv_g < 0. \end{aligned}$$

Indeed:

Set $A_{\epsilon }(p) = {B_{2\epsilon }(p)} \backslash {B_{\epsilon }(p)}$, then one has

$$\begin{aligned} \int _M (\phi _{\epsilon }v_1)P_{g}^n(\phi _{\epsilon }v_1)dv_g= & {} \underset{I_1}{ \underbrace{\int _{B_{\epsilon }(p)} (\phi _{\epsilon }v)P_{g}(\phi _{\epsilon }v)dv_g}} + \underset{I_2}{ \underbrace{\int _{A_{\epsilon }(p) } (\phi _{\epsilon }v)P_{g}(\phi _{\epsilon }v)dv_g}}\\&\quad + \underset{I_3}{ \underbrace{\int _{M \backslash {B_{2\epsilon }(p)} }(\phi _{\epsilon }v)P_{g}(\phi _{\epsilon }v)dv_g} }. \end{aligned}$$

Clearly the first integral $I_1=0$ (since $\phi _{\epsilon }=0$ on the ball $B_{\epsilon }(p)$). For the second integral $I_2$ since $v\in C^{\infty }(M)$, we can find a constant $C>0$ such that

$$\begin{aligned} |I_2 |\le & {} \int _{A_{\epsilon }(p)}(\vert \Delta _{g} ^{\frac{k}{2}}(\phi _{\epsilon }v)\vert ^{2}+ \sum \limits _{l=0}^{k-1} |A_{(l)}(g) (\nabla ^{l}\phi _{\epsilon }v, \nabla ^{l}\phi _{\epsilon }v)|) dv_{g} \nonumber \\\le & {} C\left( \int _{A_{\epsilon }(p)} (\vert \Delta _{g} ^{\frac{k}{2}}(\phi _{\epsilon })\vert ^{2}+\sum \limits _{l=0}^{k-1} \vert \nabla ^{l}(\phi _{\epsilon })\vert ^{2})dv_{g} \right) . \end{aligned}$$

(29)

The latter inequality (29) is a direct consequence of formula (22).

Using ($28_1$), ($28_2$), (29) and passing to the polar coordinates, we can easily find constants $C_k, C_{k-1},\ldots ,C_0 >0$ such that,

$$\begin{aligned} |I_2 |\le \frac{C_k}{\epsilon ^{2k}}\int _\epsilon ^{2\epsilon } r^{n-1}dr+ \left( \sum \limits _{l=0}^{k-1}\frac{C_l}{\epsilon ^{2l}}\right) \int _\epsilon ^{2\epsilon } r^{n-1}dr \underset{\epsilon \rightarrow 0}{\longrightarrow } 0 \quad \text {since} \quad n>2k \end{aligned}$$

which means that the second integral:

$$\begin{aligned} I_2 \underset{\epsilon \rightarrow 0}{\longrightarrow } 0. \end{aligned}$$

And finally since $\phi _{\epsilon }=1$ on $M\backslash B_{2\epsilon }(p)$, the third integral $I_3 = \int _{M \backslash {B_{2\epsilon }(p)} }vP_{g}vdv_g$.

This implies that:

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\int _M (\phi _{\epsilon }v)P_{g}(\phi _{\epsilon }v)dv_g =\lim _{\epsilon \rightarrow 0} (I_1+I_2+I_3) =\lim _{\epsilon \rightarrow 0}I_3=\int _M vP_{g}vdv_g < 0. \end{aligned}$$

If we put $w = \phi _{\epsilon }v$, for $\epsilon$ small enough, we still have $\int _M wP_gwdv_g < 0$.

Now, let $u_\epsilon \ge 0$ be a smooth function with support in $B_{\epsilon }(p)$ and let $\overline{g}=u_{\epsilon }^{\frac{N-2}{k}}g$ since

$$\begin{aligned} \lambda _{1}(\overline{g})=\inf _{v \in H_{k}^{2}(M), v\ne 0} \frac{\int _MvP_ {g}vdv_g}{\int _M u_\epsilon ^{N-2}v^2dv_g} \end{aligned}$$

it follows that for any real $\alpha >0$, one has

$$\begin{aligned} \lambda _{1}(\overline{g}) \leqslant \lim _{\alpha \longrightarrow 0} \frac{\int _M(w+\alpha )P_{g}(w+\alpha )dv_g}{\int _M u_\epsilon ^{N-2}(w+\alpha )^2dv_g}=-\infty . \end{aligned}$$

Indeed

$$\begin{aligned} \lim _{\alpha \rightarrow 0} \int _M u_\epsilon ^{N-2} (w+\alpha )^2 dv_g=0 \end{aligned}$$

and

$$\begin{aligned} \lim _{\alpha \rightarrow 0} (\int _M(w+\alpha )P_ {g}(w+\alpha )dv_{g} =\int _M wP_g wdv_{g}<0. \end{aligned}$$

$\square$

Theorem 3.2

Let $\overline{g} = u^{\frac{4}{n-2k}}g$ be any generalized metric to g such that $u>0$. Assume that $\lambda _{1}({g}) > 0$. Then there exists a nontrivial function v in $H_{k}^{2}(M)$ such that, in the weak sense, v satisfy :

$$\begin{aligned} P_{g}v=\lambda _{1}(\overline{g})u^{N-2}v \quad \text {and} \quad \int \nolimits _{M}u^{N-2}v^{2}dv_{g}=1 \end{aligned}$$

(30)

Moreover, if $u\in C^{\infty }(M)$, then $v \in C^{\infty }(M)$ and if (M, g) is Einstein and $S_g > 0$, the solution $v >0$.

Proof

Let $(v_{m})_m$ be a minimizing sequence for $\lambda _{1}(\overline{g})$. In other words, the sequence $(v_{m})_m \in H_{k}^{2}(M)$, $u^{\frac{N-2}{2}}v_{m}$ $\ne 0$ and such that

$$\begin{aligned} \underset{m}{\lim }\frac{\int _{M}v_mP_{g}v_mdv_{g}}{ \int _{M}u^{N-2}v_{m}^{2}dv_{g}}=\lambda _{1}(\overline{g}). \end{aligned}$$

(31)

Without loss of generality, we can always normalize $v_{m}$ by $\int _{M}u^{N-2}v_{m}^{2}dv_{g}=1$.

Since $\lambda _{1}({g}) > 0$, $P_g$ is coercive. Then Theorem (3.1) implies that the sequence $(v_{m})$ is bounded in $H_{k}^{2}(M)$, and after restriction to a subsequence we may assume that there exists v in $H_{k}^{2}(M)$ such that $v_{m}\rightarrow v$ weakly in $H_{k}^{2}(M)$, strongly in $H_{k-1}^{2}(M)$ and almost everywhere in M. Again since $P_g$ is coercive, Proposition (2.2) implies that $\Vert . \Vert _{P_g}$ is a norm on $H_{k}^{2}(M)$ equivalent to the standard norm $\Vert . \Vert _{H_{k}^2}$, then by standard argument, one has

$$\begin{aligned} \int _{M}vP_{g}vdv_{g} \le \lim \inf \int _{M}v_mP_{g}v_mdv_{g} =\lambda _{1}(\overline{g}), \end{aligned}$$

as in [6] from (Lemma (4)), we get

$$\begin{aligned} \int \nolimits _{M}u^{N-2}|v^{2}-v_{m}^{2} |\ dv_{g}\rightarrow 0\text { i.e }\int \nolimits _{M}u^{N-2}v^{2}dv_{g}=1 \end{aligned}$$

and since $\lambda _{1}(\overline{g})$ is the infimum, one gets

$$\begin{aligned} \int _{M}vP_{g}vdv_{g}=\lambda _{1}(\overline{g}). \end{aligned}$$

Consequently v is a non-trivial weak minimizer of the functional associated to $\lambda _{1}(\overline{g})$. Writing the Euler-Lagrange equation, we find that v satisfies in the weak sense the equation

$$\begin{aligned} P_{g}v=\lambda _{1}(\overline{g})u^{N-2}v. \end{aligned}$$

Moreover, since v is nontrivial, we have

$$\begin{aligned} \lambda _{1}(\overline{g})=\int _{M}vP_{g}vdv_{g}=\Vert v \Vert _{P_g}^2 >0. \end{aligned}$$

(32)

If $u\in C_{+}^{\infty }(M)$, we get $\lambda _{1}(\overline{g})u^{N-2}v \in H_{k}^{2}(M)$, then $P_{g}v \in H_{k}^{2}(M)$ and by regularity theorems $v \in H_{3k}^{2}(M),$ it follows by successive iterations that $v \in H_{l}^{2}(M)$ where l is large enough and finally if $\frac{1}{2} < \frac{l-m}{n}$, one gets

$$\begin{aligned} H_{l}^{2}(M)\subset C^{m}(M) \end{aligned}$$

so we can take $m = 2k$ i.e

$$\begin{aligned} v \in C^{2k}(M), \text {therefore} v \in C^{\infty }(M). \end{aligned}$$

In particular, if (M, g) is Einstein and $S_g > 0$, from [6] (Proposition (7)), one has

$$\begin{aligned} v > 0. \end{aligned}$$

$\square$

Remark 3.1

Let v be the solution of the Eq. (30). Then there exists a nontrivial function w in $H_{k}^{2}(M)$ such that, in the weak sense one has :

$$\begin{aligned} P_{g}w=\lambda _{2}^{\prime }(\overline{g})u^{N-2}w \end{aligned}$$

with the constraints $\int \nolimits _{M}u^{N-2}w^{2}dv_{g}=1$ and $\int \nolimits _{M}u^{N-2}vwdv_{g}=0$ where

$$\begin{aligned} \lambda _{2}^{\prime }(\overline{g})= \inf \frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}u^{N-2}v^{2}dv_{g}} \end{aligned}$$

and the infimum is taken over the set

$$\begin{aligned} E=\left\{ w \in H_{k}^{2}(M) \text {such that} u^{\frac{N-2}{2}}w \ne 0, \int \nolimits _{M}u^{N-2}w^{2}dv_{g}=1 \text {and} \int \nolimits _{M}u^{N-2}wvdv_{g}=0 \right\} . \end{aligned}$$

Proof

Let $(w_{m})_m$ be a minimizing sequence for $\lambda _{2}^{\prime }(\overline{g})$, with the same method as above, we find non trivial minimizer w to $\lambda _{2}^{\prime }(\overline{g})$ such that $P_{g}w=\lambda _{2}^{\prime }(\overline{g})u^{N-2}w$ in the weak sens with $\int _{M}u^{N-2}w^{2}dv_{g}=1$. Now writing

$$\begin{aligned} \int \nolimits _{M}u^{N-2}wvdv_{g}= & {} \int \nolimits _{M}u^{N-2}w_{m}v-u^{N-2}w_{m}v+u^{N-2}wvdv_{g}\\= & {} \int \nolimits _{M}u^{N-2}v(w-w_{m})dv_{g}+\int \nolimits _{M}u^{N-2}w_{m}vdv_{g}=0. \end{aligned}$$

As the sequence $w_{m}\in E$, $\int \nolimits _{M}u^{N-2}w_{m}vdv_{g}=0$, and by using the weak convergence of $w_{m}$ to w in $L^{N}(M)$ and since $u^{N-2}v \in L^{\frac{N}{N-1}}(M)$ where $L^{\frac{N}{N-1}} (M)$ is the dual space of $L^{N}(M)$, we get $\int \nolimits _{M}u^{N-2}v(w-w_{m})dv_{g}\rightarrow 0$ thus,

$$\begin{aligned} \int \nolimits _{M}u^{N-2}wvdv_{g}=0. \end{aligned}$$

If $u\in C_{+}^{\infty }(M)$, one also gets $w \in C^{2k}(M)$ and finally, as in [6] it follows that $\lambda _{2}^{\prime }(\overline{g})=\lambda _{2}(\overline{g})$. $\square$

Proposition 3.2

Let (M, g) be a compact Riemannian manifold of dimension $n\ge 3$. Assume that $\overline{g}$ is a conformal metric and $\lambda _{1}(\overline{g}) > 0$. If $Q_{g} \le 0$, then the solution v of (30) is nodal.

Proof

By Theorem (3.2), v satisfies the equation $P_{g}v=\lambda _{1}(\overline{g})u^{N-2}v$,

then from (10), one can write

$$\begin{aligned} \Delta _{g}^{k}v+\sum \limits _{l=1}^{k-1} A_{l,g}(v)+\frac{n-2k}{2}Q_{g} v =\lambda _{1}(\overline{g})u^{N-2}v \end{aligned}$$

Integrating over M, we get that

$$\begin{aligned} \int \nolimits _{M}\Delta _{g}^{k}vdv_g +\sum \limits _{l=1}^{k-1}\int \nolimits _{M} A_{l,g}(v)dv_g+ \int \nolimits _{M}\frac{n-2k}{2}Q_{g}vdv_g =\lambda _{1}(\overline{g})\int \nolimits _{M}u^{N-2}vdv_g. \end{aligned}$$

Since $\overline{g}$ is conformal, again from Theorem (3.2), $v \in C^{\infty }(M)$ and this implies that

$$\begin{aligned} \underset{= 0}{\underbrace{\int \nolimits _{M}\Delta _{g}^{k}vdv_g}}+ \underset{= 0}{\underbrace{ \sum \limits _{l=1}^{k-1}\int \nolimits _{M} A_{l,g}(v) dv_g } } +\int \nolimits _{M}\frac{n-2k}{2}Q_{g}vdv_g =\lambda _{1}(\overline{g})\int \nolimits _{M}u^{N-2}vdv_g. \end{aligned}$$

Since $\lambda _{1}(\overline{g}) > 0$ and $Q_{g} \le 0$, hence if $v\ge 0$, one has

$$\begin{aligned} \underset{\le 0}{\underbrace{ \int _{M}Q_{g}vdv_{g}}}\text { }=\text { } \underset{ > 0}{\underbrace{\int _{M}\lambda _{1}(\overline{g})u^{N-2}vdv_{g}}} \end{aligned}$$

this makes a contradiction, if $v\le 0$, one has

$$\begin{aligned} \underset{\ge 0}{\underbrace{\int _{M}Q_{g}vdv_{g}}}=\underset{< 0 }{\underbrace{\int _{M}\lambda _{1}(\overline{g})u^{N-2}vdv_{g}}} \end{aligned}$$

and this is also a contradiction. Consequently, v changes the sign.

If $\lambda _{1}(\overline{g}) < 0$ and $Q_{g} \ge 0$. With the same method, we get the same thing.

$\square$

4 Existence of a minimum of $\mu _{1}$

In this section, we study the first GJMS invariant $\mu _{1}$ in case $\lambda _{1}(g) > 0$. We will prove that $\mu _{1}$ is attained by a generalized metric. However, if $\lambda _{1}({g})<0$, Proposition (3.1) implies that $\mu _{1}$ is not well defined. In other words, from the variational characterization of $\mu _{1}$, one has

$$\begin{aligned} \mu _{1}=-\infty . \end{aligned}$$

In order to prove Theorem (4.1), we prove some useful lemmas.

Definition 4.1

In this definition, we precise the formula (9). Indeed, by using the definition of $\lambda _{1}(\overline{g})$ formula (8), the first GJMS invariant $\mu _{1}$ is given by

$$\begin{aligned} \mu _{1}= & {} \underset{\overline{g}\in [g]}{\inf }\lambda _{1}(\overline{g}) Vol(M,\overline{g})^{^{\frac{2k}{n}}}\\= & {} \underset{ \underset{V \in Gr_{1}^{u}(H_{k}^{2}(M))}{u\in C_{+}^{\infty }(M) }}{\inf } \underset{v \in V^*}{ \sup }\frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}u^{N-2}v^{2}dv_{g}}\left( \int \nolimits _{M}u^{N}dv_{g}\right) ^{\frac{2k}{n}}. \end{aligned}$$

Lemma 4.1

We have:

$$\begin{aligned} \mu _{1}\le \mu \end{aligned}$$

(33)

where $\mu$ is the GJMS invariant, see (4).

Proof

$$\begin{aligned} \mu _{1}= & {} \underset{\overline{g}\in [g]}{\inf }\lambda _{1}(\overline{g})Vol(M,\overline{g})^{^{\frac{2k}{n}}}\\= & {} \underset{\overline{g}\in [g]}{\inf }\lambda _{1}(u^{\frac{N-2}{k}}g)Vol(M,\overline{g})^{^{\frac{2k}{n}}}\\= & {} \underset{u\in C_{+}^{\infty }(M)}{\inf }\left( \underset{V\in Gr_{1}^{u}(H_{k}^{2}(M))}{\inf }\underset{v \in V^*}{\sup }\frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}u^{N-2}v^{2}dv_{g}}\right) \left( \int \nolimits _{M}u^{N}dv_{g}\right) ^{\frac{2k}{n}} \end{aligned}$$

where $V^*=V \backslash \lbrace 0 \rbrace$.

From the embedding $C_{+}^{\infty }(M)\subset H_{k}^{2}(M)$, one can write

$$\begin{aligned} \mu _{1} \le \underset{ \underset{V \in Gr_{1}^{u}(C_{+}^{\infty }(M))}{u\in C_{+}^{\infty }(M) }}{\inf } \underset{v\in V^*}{\text { }\sup }\frac{\int _{M}vP_{g}vdv_{g}}{ \int _{M}u^{N-2}v^{2}dv_{g}}\left( \int \nolimits _{M}u^{N}dv_{g}\right) ^{\frac{2k}{n}} \end{aligned}$$

in particular for $u=v$, one has

$$\begin{aligned} \mu _{1}\le & {} \underset{ \underset{V \in Gr_{1}^{u}(C_{+}^{\infty }(M))}{u\in C_{+}^{\infty }(M) }}{\inf }\underset{v\in V^*}{\text { }\sup }\frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}v^{N-2}v^{2}dv_{g}}\left( \int \nolimits _{M}v^{N}dv_{g}\right) ^{\frac{2k}{n}}\\\le & {} \underset{ \underset{V \in Gr_{1}^{u}(C_{+}^{\infty }(M))}{u\in C_{+}^{\infty }(M) }}{\inf }\underset{v\in V^*}{\text { }\sup }\frac{\int _{M}vP_{g}vdv_{g}}{(\int \nolimits _{M}v^{N}dv_{g})^{1-\frac{2k}{n}}}\\\le & {} \underset{ \underset{V \in Gr_{1}^{u}(C_{+}^{\infty }(M))}{u\in C_{+}^{\infty }(M) }}{\inf }\underset{v\in V^*}{\text { }\sup }\frac{\int _{M}vP_{g}vdv_{g}}{(\int \nolimits _{M}v^{N}dv_{g})^{\frac{2}{N}}}. \end{aligned}$$

Since $V \in Gr_{1}^{u}(C_{+}^{\infty }(M)$, $V = \lbrace \lambda v , \lambda \in {\mathbb {R}}^* \rbrace$ where $v \in C_{+}^{\infty }(M)$, then we deduce that:

$$\begin{aligned} \underset{v\in V^*}{\text { }\sup }\frac{\int _{M}vP_{g}vdv_{g}}{ (\int \nolimits _{M}v^{N}dv_{g})^{\frac{2}{N}}} =\underset{\lambda \in {\mathbb {R}}^* }{\text { }\sup }\frac{\int _{M}(\lambda v)P_{g}(\lambda v)dv_{g}}{ \left( \int \nolimits _{M}(\lambda v)^{N}dv_{g}\right) ^{\frac{2}{N}}} =\frac{\int _{M}vP_{g}vdv_{g}}{ (\int \nolimits _{M}v^{N}dv_{g})^{\frac{2}{N}}}. \end{aligned}$$

This implies that

$$\begin{aligned} \mu _{1}\le & {} \underset{v\in C_{+}^{\infty }(M)}{ \inf }\frac{\int _{M}vP_{g}vdv_{g}}{ \left( \int \nolimits _{M}v^{N}dv_{g}\right) ^{\frac{2}{N}}}= \mu . \end{aligned}$$

$\square$

Lemma 4.2

Let $(v_{m})$ and $(u_{m})$ be two sequences such that $v_{m}\rightarrow v$ weakly in $\ H_{k}^{2}(M)$, $\ u_{m}$ $\rightarrow u$ weakly in $L^{N}(M)$ and checking $\int _{M}u_{m}^{N-2}{}v_{m}^{2}{}dv_{g}=1$. Then

$$\begin{aligned} \int _{M}u_{m}^{N-2}(v_{m}-v)^{2}dv_{g}=1-\int _{M}u^{N-2}v^{2}dv_{g}+o(1). \end{aligned}$$

Proof

Writing

$$\begin{aligned} \int _{M}u_{m}^{N-2}(v_{m}-v)^{2}dv_{g}= & {} \int _{M}u_{m}^{N-2}v_{m}{}^{2}dv_{g}+ \int _{M}u_{m}^{N-2}v^{2}dv_{g}-\int _{M}2u_{m}^{N-2}v_{m}vdv_{g}\\= & {} 1+\int _{M}u_{m}^{N-2}v^{2}dv_{g}-\int _{M}2u_{m}^{N-2}v_{m}vdv_{g}. \end{aligned}$$

The sequence $u_{m}^{N-2}$ is bounded in $L^{\frac{N}{N-2}}(M)$ and converges almost everywhere to $u^{N-2}$ on M, hence $u_{m}^{N-2}\rightarrow u^{N-2}$ weakly in $L^{\frac{N}{N-2}}(M)$.

This means that for all $\phi$ in $L^{\frac{N}{2}}(M)$, one gets $\int _{M} u_{m}^{N-2}\phi dv_{g}\rightarrow \int _{M} u^{N-2}\phi dv_{g}$.

In particular for $\phi =v^{2}$, we obtain

$$\begin{aligned} \int _{M}u_{m}^{N-2}v^{2}dv_{g}\rightarrow \int _{M}u^{N-2}v^{2}dv_{g}. \end{aligned}$$

On the other hand since

$$\begin{aligned} \int _{M}u_{m}^{N-2}{}^{\frac{N}{N-1}}v_{m}^{\frac{N}{N-1}}dv_{g}\le \left( \int _{M}u_{m}{}^{N}dv_{g}\right) ^{\frac{N-2}{N-1}}\left( \int _{M}v_{m}^{N}dv_{g}\right) ^{ \frac{1}{N-1}} \end{aligned}$$

this means that the sequence $u_{m}^{N-2}v_{m}$ is also bounded in $L^{\frac{N}{N-1}}(M)$ and since $u_{m}^{N-2}v_{m}$ goes to $u{}^{N-2}v$ almost everywhere, one has $u_{m}^{N-2}v_{m}\rightarrow u{}^{N-2}v$ weakly in $L^{\frac{N}{N-1}}(M)$, then for all $\phi \in L^{N}(M)$, one has $\int _{M}u_{m}^{N-2}v_{m}\phi dv_{g}\rightarrow \int _{M}u{}^{N-2}v \phi dv_{g}$. In particular for $\phi =v \in L^{N}(M)$, we obtain

$$\begin{aligned} \int _{M}u_{m}^{N-2}v_{m}vdv_{g}\rightarrow \int _{M}u{}^{N-2}v^{2}dv_{g} \end{aligned}$$

Consequently,

$$\begin{aligned} \int _{M}u_{m}^{N-2}(v_{m}-v)^{2}dv_{g}=1-\int _{M}u^{N-2}v^{2}dv_{g}+o(1). \end{aligned}$$

(34)

$\square$

Theorem 4.1

Let (M, g) be a compact Riemannian manifold of dimension $n \ge 3$. Assume that $\lambda _{1}(g) > 0$ and

$$\begin{aligned} 1- \mu K_{0}>0 \end{aligned}$$

(35)

where $\mu$ is the GJMS invariant and $K_{0}$ is given by (20). Then there exist two nontrivial functions $u \in L_+^{N}(M)$ and $v \in H_{k}^{2}(M)$ such that in the weak sense, we have

$$\begin{aligned} P_{g}v=\mu _{1}u^{N-2}v \quad and \quad \int _{M}u^{N-2}v^{2}dv_{g}=1. \end{aligned}$$

(36)

In other words, $\mu _{1}$ is attained by a generalized metric.

Proof

let $g_{m}=u_{m}^{\frac{4}{n-2k}}g$ be a minimizing sequence of conformal metrics of $\mu _{1}$, a sequence of metrics such that $u_{m}\in C^{\infty }(M),$ $u_{m}>0$ and

$$\begin{aligned} \mu _{1}= \underset{m}{\lim } \lambda _{1}(g_m) vol(M,g_{m})^{\frac{2k}{n}} \end{aligned}$$

For more clarity we set : $\lambda _{1}(g_m)=\lambda _{1,m}$.

Without loss of generality, we may assume that

$$\begin{aligned} vol(M,g_{m})=\int \nolimits _{M}u_{m}^{N}dv_{g}=1. \end{aligned}$$

(37)

Indeed, since

$$\begin{aligned} \frac{2kN}{n} =\frac{2k2n}{n(n-2k)}=\frac{2n}{n-2k}-2=N-2, \end{aligned}$$

it follows that for any $\lambda > 0$, one gets

$$\begin{aligned} I(\lambda u,v) =\frac{\int _{M}vP_{g}vdv_{g}}{\int _{M} (\lambda u )^{N-2}v^{2}dv_{g}}\left( \int \nolimits _{M} (\lambda u)^{N}dv_{g}\right) ^{\frac{2k}{n}}=I(u,v). \end{aligned}$$

This means that if $(u_{m})$ is a minimizing sequence, $(\lambda u_{m})_{m}$ is also is a minimizing sequence, just choose $\lambda = (\int \nolimits _{M}u_{m}^{N}dv_{g})^{-\frac{1}{N}}.$ i.e

$$\begin{aligned} \mu _{1}= \underset{m}{ \lim }\lambda _{1,m}. \end{aligned}$$

Step 1: Firstly, (37) implies that the sequence $(u_{m})_m$ is bounded in $L^{N}(M)$, hence there exists $u \in L^{N}(M)$, $u \ge 0$ such that $u_{m} \rightarrow u$ weakly in $L^{N}(M)$ and by standard argument, we get

$$\begin{aligned} \int \nolimits _{M}u^{N}dv_{g} \le \lim \inf \int \nolimits _{M}u_{m}^{N}dv_{g}=1. \end{aligned}$$

(38)

Now, we are going to prove that the generalized metric $u^{\frac{4}{n-2k}}g$ with $u\in L^{N}(M)$, $u\ge 0$ and $u\ne 0$ minimizes $\mu _{1}$.

Since $\lambda _{1}(g) > 0$, $P_{g}$ is coercive. Then for all $u_{m}\in C^{\infty }(M)$, Theorem(3.2) implies the existence of $v_{m}\in C^{\infty }(M)$ such that

$$\begin{aligned} P_{g}v_{m}=\lambda _{1,m}u_{m}^{N-2}v_{m}\text { and } \int _{M}u_{m}^{N-2}v_{m}^{2}dv_{g}=1. \end{aligned}$$

Now for m large enough, we may assume that

$$\begin{aligned} \lambda _{1,m} \le \mu _{1}+1 \end{aligned}$$

which implies that

$$\begin{aligned} \Vert v_{m}\Vert _{P_g}^{2}=\int _{M}v_{m}P_{g}v_{m}dv_{g}=\lambda _{1,m}\le \mu _{1}+1. \end{aligned}$$

Hence the sequence $(v_{m})_m$ is bounded in $H_{k}^{2}(M)$, then there exists $v\in H_{k}^{2}(M)$ such that $v_m \longrightarrow v$ weakly in ${H_{k}^{2}}(M)$ and $v_m \longrightarrow v$ strongly in $H_{k-1}^{2}(M)$. This, together with the weak convergence of $(u_m)_m$, imply that the function v is a weak solution of the following equation

$$\begin{aligned} P_{g}v=\mu _{1}u^{N-2}v. \end{aligned}$$

(39)

Step 2: we show that u, v are not identically null.

Letting $\varphi _m=v_{m}-v$ and

$$\begin{aligned} A =\int _{M} u_{m}^{N-2}\varphi _m^{2}dv_{g}. \end{aligned}$$

(40)

Clearly $\varphi _m \rightarrow 0$ and the strong convergence of $\varphi _m$ in $H_{k-1}^{2}(M)$ implies that

$$\begin{aligned} \int _{M}\sum \limits _{l=0}^{k-1}\vert \nabla ^{l}(\varphi _m) \vert ^{2}dv_{g}=o(1) \quad \text {and} \quad \sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}\varphi _m, \nabla ^{l}\varphi _m)dv_{g}=o(1). \end{aligned}$$

Then by Hölder inequality, Theorem (2.2) and Brezis-Lieb lemma, one has

$$\begin{aligned} A\le & {} \left( \int _{M} (u_{m})^{N-2 \frac{N}{N-2}}dv_{g}\right) ^{\frac{N-2}{N}}\left( \int _{M}(\varphi _m\right) ^{\frac{N}{2}2}dv_{g}) ^{\frac{2}{N}} \\\le & {} \Vert \varphi _m\Vert _{N}^{2}\\\le & {} (K_{0}+\varepsilon ) \int \nolimits _{M}\vert \Delta _{g}^{\frac{k}{2}}\varphi _m \vert ^{2}dv_{g}+ B_{\varepsilon }\int _{M}\sum \limits _{l=0}^{k-1}\vert \nabla ^{l}\varphi _m \vert ^{2}dv_{g}\\\le & {} (K_{0}+\varepsilon )\int _{M}\vert \Delta _{g} ^{\frac{k}{2}}(v_{m})\vert ^{2}-\vert \Delta _{g} ^{\frac{k}{2}}(v)\vert ^{2}dv_{g}+o(1). \end{aligned}$$

Therefore,

$$\begin{aligned}&A\le (K_{0}+\varepsilon ) \int \nolimits _{M}(\vert \Delta _{g}^{ \frac{k}{2}}(v_{m})\vert ^{2}- \vert \Delta _{g} ^{\frac{k}{2}}(v)\vert ^{2})dv_{g}\\&\quad +\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}\varphi _m, \nabla ^{l}\varphi _m)dv_{g}+o(1). \end{aligned}$$

Again from the strong convergence in $H_{k-1}^{2}(M),$ one also gets

$$\begin{aligned}&\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}\varphi _m, \nabla ^{l}\varphi _m)dv_{g}=\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v_m, \nabla ^{l}v_m)dv_{g}\\&\quad -\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v, \nabla ^{l}v)dv_{g}+o(1), \end{aligned}$$

then

$$\begin{aligned} A\le & {} (K_{0}+\varepsilon ) \left[ \int \nolimits _{M}(\vert \Delta _{g}^{\frac{k}{2}}(v_{m})\vert ^{2}-\vert \Delta _{g} ^{\frac{k}{2}}(v)\vert ^{2})dv_{g}\right. \\+ & {} \left. \sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v_m, \nabla ^{l}v_m)dv_{g}-\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v, \nabla ^{l}v)dv_{g} \right] + o(1). \end{aligned}$$

Since

$$\begin{aligned} \int _{M}vP_{g}vdv_{g}=\int \nolimits _{M}(\vert \Delta _{g} ^{\frac{k}{2}}(v) \vert ^{2}+\sum \limits _{l=0}^{k-1}\int \nolimits _{M} A_{(l)}(g) (\nabla ^{l}v, \nabla ^{l}v)dv_{g}, \end{aligned}$$

We deduce that

$$\begin{aligned} A\le & {} (K_{0}+\varepsilon )\left( \int _{M}v_{m}P_{g}v_{m}-vP_{g} vdv_{g}\right) + o(1)\\\le & {} (K_{0}+\varepsilon )\left( \lambda _{1,m}-\int _{M}vP_{g}vdv_{g}\right) +o(1)\\\le & {} (K_{0}+\varepsilon )\left( \lambda _{1,m}-\mu _{1} \int _{M}u^{N-2}v^{2}dv_{g})\right) +o(1) \end{aligned}$$

Independently, with Lemma (4.2) formula (34), we have

$$\begin{aligned} A = 1-\int _{M}u^{N-2}v^{2}dv_{g}+o(1) \end{aligned}$$

then it follows that

$$\begin{aligned} 1-\int _{M}u^{N-2}v^{2}dv_{g}\le & {} (K_{0}+\varepsilon )\left( \lambda _{1,m}-\mu _{1} \int _{M}u^{N-2}v^{2}dv_{g})\right) +o(1). \end{aligned}$$

Now when $m \rightarrow +\infty$, one gets

$$\begin{aligned} 1-\int _{M}u^{N-2}v^{2}dv_{g}\le (K_{0}+\varepsilon )(\mu _{1}-\mu _{1}\int _{M}u^{N-2}v^{2}dv_{g}) \end{aligned}$$

therefore,

$$\begin{aligned} 1-(K_{0}+\varepsilon ) \mu _{1} \le \int _{M}u^{N-2}v^{2}dv_{g}-(K_{0}+\varepsilon )\mu _{1}\int _{M}u^{N-2}v^{2}dv_{g} \end{aligned}$$

and this leads that

$$\begin{aligned} 1-K_{0}\mu _{1} \le (1-K_{0}\mu _{1})\int _{M}u^{N-2}v^{2}dv_{g}+\varepsilon \mu _{1}(1-\int _{M}u^{N-2}v^{2}dv_{g}). \end{aligned}$$

(41)

Now, by using Lemma (4.1) formula (33) and the assumption (35), one easily has $1-K_{0}\mu _{1}>0$ and as we can choose $\varepsilon$ sufficiently small enough, (41) necessarily implies that

$$\begin{aligned} \int _{M}u^{N-2}v^{2}dv_{g}\ge 1. \end{aligned}$$

Fatou’s lemma, implies that

$$\begin{aligned} \int _{M}u^{N-2}v^{2}dv_{g}\le \lim \inf \int _{M}u_{m}^{N-2}v_{m}^{2}dv_{g}=1 \end{aligned}$$

then we deduce that

$$\begin{aligned} \int _{M}u^{N-2}v^{2}dv_{g}=1. \end{aligned}$$

This implies that v and u are not identically null which means that $\mu _{1}$ is attained by the generalized metric $u^{\frac{4}{n-2k}}g$. Moreover, we obtain

$$\begin{aligned} \mu _{1}=\Vert v\Vert _{g}^{2}=\int _{M}vP_{g}vdv_{g}>0. \end{aligned}$$

(42)

$\square$

5 Nonlinear GJMS equation and nodal solution

In this section, we show that the equation $P_{g}v=\mu _{1}\vert v \vert ^{N-2} v$ has a nodal solution if $Q_{g}\le 0$.

Theorem 5.1

Let (M, g) be a compact Riemannian manifold of dimension $n \ge 3$. Assume that $\mu _{1}$ is attained by the generalized metric $u^{\frac{4}{n-2k}}g$ where $u \in L_+^{N}(M)$. Then $u = \vert v \vert$ where $v \in H_{k}^{2}(M)$, v is a solution weak of $P_{g}v=\mu _{1}u^{N-2}v$ and such that $\int _{M}u^{N-2}v^{2}dv_{g}=1$. Moreover, the function $v \in C^{2k}(M)$ and if $Q_{g}\le 0$, then v changes the sign.

Proof

Let the function $h=a|v |\in L_{+}^{N}(M)$ with $a>0$ chosen such that $\int _{M}h^{N}dv_{g}=1$, by definition

$$\begin{aligned}&\mu _{1}\le \frac{\int _{M}vP_{g}vdv_{g}}{\int _{M}h^{N-2}v^{2}dv_{g}}=\frac{ \mu _{1}\int _{M}u^{N-2}v^{2}dv_{g}}{\int _{M}h^{N-2}v^{2}dv_{g}}=\frac{a^{2} \mu _{1}\int _{M}u^{N-2}v^{2}dv_{g}}{a^{2}\int _{M}h^{N-2}v^{2}dv_{g}}\\&=\frac{\mu _{1}\int _{M}u^{N-2}(av)^{2}dv_{g}}{\int _{M}(a|v |)^{N-2}(av)^{2}dv_{g}}=\frac{ \mu _{1}\int _{M}u^{N-2}(av)^{2}dv_{g}}{\int _{M}h^{N}dv_{g}}= \mu _{1}\int _{M}u^{N-2}(a |v |)^{2}dv_{g}. \end{aligned}$$

By using (38) and H$\ddot{o}$lder’s inequality, it follows that

$$\begin{aligned} \mu _{1}\le & {} \mu _{1}\left( \int _{M}u^{N-2\frac{N}{N-2}}dv_g\right) ^{\frac{N-2}{N}}\left( \int _{M}(a |v |)^{2\frac{N}{2}}dv_{g}\right) ^{\frac{2}{N}}\nonumber \\\le & {} \mu _{1}\left( \int _{M}u^{N}dv_g\right) ^{\frac{N-2}{N}}\left( \int _{M}h^{N}dv_{g}\right) ^{\frac{2}{N}}\nonumber \\\le & {} \mu _{1}\left( \int _{M}u^{N}dv_g\right) ^{\frac{N-2}{N}} \le \mu _{1}, \end{aligned}$$

(43)

this implies that we have both equality in the H$\ddot{o}$lder inequality. The equality in the H$\ddot{o}$lder inequality implies that there exists a constant $b>0$ such that :

$$\begin{aligned} u=b|v |. \end{aligned}$$

From the equality $1= \int _{M}u^{N-2}v^{2}dv_{g}=b^{N-2}\int _{M}|v |^{N}dv_{g},$ we obtain

$$\begin{aligned} \frac{1}{b^{N-2}}=\int _{M}|v |^{N}dv_{g}. \end{aligned}$$

(43) implies that $\int _{M}u^{N}dv_g=1$, then it follows that

$$\begin{aligned} b^{N}\int _{M}|v |^{N}dv_{g}= 1 \end{aligned}$$

which leads to

$$\begin{aligned} \frac{1}{b^{N-2}}=\int _{M}|v |^{N}dv_{g} = \frac{1}{b^{N}}. \end{aligned}$$

Therefore

$$\begin{aligned} b = 1 \quad \text {and} \quad u=|v |. \end{aligned}$$

Hence, v is a weak solution of

$$\begin{aligned} P_{g}v=\mu _{1}\vert v \vert ^{N-2} v \end{aligned}$$

(44)

and from standard regularity see Proposition(2.5), we get that $v \in C^{2k}(M)$. In addition, since $\mu _{1} > 0$ and $Q_{g} \le 0$, by following the same proof of Proposition (3.2), we deduce that the function v changes the sign. $\square$

6 Case of Einsteinian manifold and positive solution

In this section, on Einstein manifold when $S_g > 0$, we will prove that the solution v of Eq. (44) is positive, $\mu _{1}=\mu$ and is attained by a conformal metric which leads to the existence of a metric $\overline{g}$ conformal to g such that the Q-curvature is constant. In the case $S_{g} < 0$, and k is odd, the solution is nodal.

Theorem 6.1

Let (M, g) be a compact Einstein manifold of dimension $n \ge 3$. Assume that $S_g > 0$ and $1- \mu K_{0}>0$ where $\mu$ is the GJMS invariant and $K_{0}$ is given by formula (20). Then $\mu _{1}$ is attained by the conformal metric $u ^{\frac{4}{n-2k}}g$. In other words, there exists $u \in C^{\infty }(M), u>0$ solution to the following equation

$$\begin{aligned} P_{g}u=\mu _{1}u^{N-1} \quad \text {such that} \quad \int _{M}v^{N}dv_{g}=1. \end{aligned}$$

Proof

We follow the same proof of Theorem (4.1).

Let $g_{m}=u_{m}^{\frac{4}{n-2k}}g$ be a minimizing sequence of conformal metrics of $\mu _{1}$, a sequence of metrics such that $u_{m}\in C^{\infty }(M),$ $u_{m}>0$ and

$$\begin{aligned} \mu _{1}= \underset{m}{ \lim } \lambda _{1,m} \quad \text {and} \quad \int \nolimits _{M}u_{m}^{N}dv_{g}=1. \end{aligned}$$

(45)

Firstly, (45) implies that the sequence $(u_{m})_m$ is bounded in $L^{N}(M)$, hence there exists $u \in L^{N}(M)$, $u \ge 0$ such that $u_{m} \rightarrow u$ weakly in $L^{N}(M)$.

Since (M, g) is Einstein and $S_g > 0$, $P_{g}$ is coercive. Then for all $u_{m}\in C^{\infty }(M)$, Theorem(3.2) implies the existence of $v_{m}\in C^{\infty }(M)$ such that $v_{m} > 0$ and

$$\begin{aligned} P_{g}v_{m}=\lambda _{1,m}u_{m}^{N-2}v_{m} \quad \text {and} \quad \int _{M}u_{m}^{N-2}v_{m}^{2}dv_{g}=1. \end{aligned}$$

Now for m large enough, we may assume that

$$\begin{aligned} \lambda _{1,m} \le \mu _{1}+1 \end{aligned}$$

which implies that

$$\begin{aligned} \Vert v_{m}\Vert _{P_g}^{2}=\int _{M}v_{m}P_{g}(v_{m})dv_{g}=\lambda _{1,m}\le \mu _{1}+1. \end{aligned}$$

Hence the sequence $(v_{m})_m$ is bounded in $H_{k}^{2}(M)$, then there exists $v\in H_{k}^{2}(M)$ such that $v \ge 0$, $v_m \longrightarrow v$ weakly in ${H_{k}^{2}}(M)$ and $v_m \longrightarrow v$ strongly in $H_{k-1}^{2}(M)$. This, together with the weak convergence of $(u_m)_m$, imply that the function v is a weak solution of the following equation

$$\begin{aligned} P_{g}v=\mu _{1}u^{N-2}v. \end{aligned}$$

(46)

and in particular

$$\begin{aligned} v \ge 0. \end{aligned}$$

Since $1-\mu K_{0} > 0$, by step (2) of the poof of Theorem (4.1), the functions u, v satisfy $\int _{M}u^{N-2}v^{2}dv_{g}=1$ and are not identically null. Since $v \ge 0$, we let the function $h=av\in L_{+}^{N}(M)$ where $a>0$ chosen such that $\int _{M}h^{N}dv_{g}=1$ and by following the same proof of Theorem (5.1), one has

$$\begin{aligned} u=v. \end{aligned}$$

Therefore, v is a weak solution of

$$\begin{aligned} P_{g}v=\mu _{1}v^{N-1} \end{aligned}$$

and from standard regularity see Proposition (2.5), we get that $v \in C^{2k}(M)$. In particular since $v \ge 0$ and $\mu _{1} > 0$, one has $P_g v \ge 0$ and since $v \ne 0$, it follows from Proposition (2.4) that $v > 0$ and again by regularity $v \in C^{\infty }(M)$.

Now since $\int _M vP_{g}vdv_g= \mu _1$, $\int \nolimits _{M}|v |^{N}dv_{g}=1$ and from the definition of $\mu$, one has

$$\begin{aligned} \mu \le \frac{\int \nolimits _{M}vP_{g}vdv_{g}}{(\int \nolimits _{M}|v |^{N}dv_{g})^{ \frac{2}{N}}}=\mu _{1}. \end{aligned}$$

(47)

It follows that

$$\begin{aligned} \mu \le \mu _{1}, \end{aligned}$$

and by Lemma (4.1) formula (33), we get that

$$\begin{aligned} \mu _{1}=\mu . \end{aligned}$$

Therefore, the infimum $\mu _1$ is achieved by the conformal metric $\overline{g}=u ^{\frac{4}{n-2k}}g$ and this means that metric $\overline{g}$ is such that the Q-curvature

$$\begin{aligned} Q_{\overline{g}}= \frac{2}{n-2k}\mu _1. \end{aligned}$$

$\square$

A more interesting situation on Einstein manifold is when $S_{g} < 0$, this implies that $\int _M vP_{g}vdv g$ can be negative or positive and consequently the eigenvalues follow same thing contrary to the case $S_g > 0$ which implies only the positivity of eigenvalues.

Corollary 6.1

Let (M, g) be a smooth compact Einstein manifold of dimension $n\ge 3$, assume that $S_{g} < 0$, $\lambda _{1}(g) > 0$ and $1- \mu K_{0}>0$. If k is odd, the following equation

$$\begin{aligned} P_{g}v=\mu _{1}|v |^{N-2}v \end{aligned}$$

(48)

has a nodal solution $v \in C^{2k}(M)$.

Proof

Since $\lambda _{1}(g) > 0$ and $1- \mu K_{0}>0$, Theorem (4.1) implies that $\mu _{1}$ is attained by a generalized metric and since (M, g) is Einstein, by using (18), (48) can be written as

$$\begin{aligned} \Delta _{g}^{k}u+\sum \limits _{l=0}^{k-2}b_{k-l-1}(S_g)^{l+1} \Delta _g^{k-l-1}u+ b_0 (S_g)^{k}u= \mu _{1}u^{N-2}v \end{aligned}$$

where $b_{k-1},\ldots , b_1, b_0$ are positive real numbers. Therefore, if k is odd, $b_0 (S_g)^{k} < 0$ and by applying Theorem (5.1) with $\frac{n-2k}{2}Q_{g} = b_0 (S_g)^{k}$, we get the result. $\square$

Proposition 6.1

Let $(\mathbb {S}^n, h)$ be the standard unit n-sphere of $\mathbb {R}^{n+1}$. Then the GJMS invariant of $\mathbb {S}^n$ is such that

$$\begin{aligned} \mu (\mathbb {S}^n)=\underset{u \in C^{\infty }(\mathbb {S}^n),u \ne 0}{\inf }\frac{\int \nolimits _{\mathbb {S}^n}uP_{{h}}(u)dv_{{h}}}{(\int \nolimits _{\mathbb {S}^n}{\vert u \vert }^{N}dv_{{h}})^{\frac{2}{N}}} = {K_{0}}^{-1}. \end{aligned}$$

Proof

We follow the same proof of Proposition (1.1) in [10]. Just note here that the choice of functions $\varphi$ and $\phi _ {\epsilon }$ must be adapted to our context, thus $\varphi \in H_{k}^{2}(M)$ and is chosen such that

$$\begin{aligned} \varphi (x)=\left( \frac{1+\vert x \vert ^2}{2}\right) ^{k-\frac{n}{2}} \end{aligned}$$

and $\phi _ {\epsilon }$ is given in Proposition (3.1).

Indeed, let $x_0$ be some point on $S^n$, and let $\phi : \mathbb {S}^n \backslash \lbrace x_0 \rbrace \longrightarrow \mathbb {R}^n$ be the stereographic projection of pole $x_0$. If $\delta$ stands for the Euclidean metric of $\mathbb {R}^n$, then

$$\begin{aligned} (\phi ^{-1})^*h=\varphi ^{\frac{N-2}{k}}\delta . \end{aligned}$$

By conformal invariance of $P_g$, we get that for all $u \in D_{k}^{2}(\mathbb {R}^{n})$,

$$\begin{aligned} \frac{\int \nolimits _{\mathbb {R}^{n}}uP_{\tilde{h}}(u)dv_{\tilde{h}}}{(\int \nolimits _{\mathbb {R}^{n}}{\vert u \vert }^{N}dv_{\tilde{h}})^{\frac{2}{N}}} = \frac{\int \nolimits _{\mathbb {R}^{n}}\vert \Delta ^{\frac{k}{2}}(u \varphi )\vert ^2 dx}{(\int \limits _{\mathbb {R}^{n}} {\vert u \varphi \vert }^{N}dx)^{\frac{2}{N}}} \end{aligned}$$

(49)

where $D_{k}^{2}(\mathbb {R}^{n})$ is given in Definition(2.4) and $\tilde{h}=(\phi ^{-1})^*h$. Suppose now that

$$\begin{aligned} \underset{u \in C^{\infty }(\mathbb {S}^n),u \ne 0}{\inf }\frac{\int \nolimits _{S^{n}}uP_{{h}}(u)dv_{{h}}}{(\int \nolimits _{\mathbb {S}^n}{\vert u \vert }^{N}dv_{{h}})^{\frac{2}{N}}} < {K_{0}}^{-1} \end{aligned}$$

(50)

and let $u_0 \in C^\infty (\mathbb {S}^n)$, $u_0\ne 0$, be such that

$$\begin{aligned} \frac{\int \nolimits _{S^{n}}u_0P_{{h}}(u_0)dv_{{h}}}{(\int \nolimits _{\mathbb {S}^n}{\vert u_0 \vert }^{N}dv_{{h}})^{\frac{2}{N}}} < {K_{0}}^{-1}. \end{aligned}$$

Fix a point p in $\mathbb {S}^n$. For $\epsilon >0$, let $\phi _ {\epsilon }$ be cut-off function i.e. a family of smooth functions on $\mathbb {S}^n$ such that :

$$\begin{aligned} \left\{ \begin{array}{l} 0 \le \phi _{\epsilon }\le 1 \\ \phi _{\epsilon }=0 \quad \text {on} \quad B_{\epsilon }(p) \quad \text {and} \quad \phi _{\epsilon }=1 \quad \text {on} \quad \mathbb {S}^{n}\backslash B_{2\epsilon }(p)\\ \vert \nabla ^l\phi _{\epsilon }\vert \le \frac{c_l}{\epsilon ^l } \quad \text {for all} \quad l \in \lbrace 1,2, ...,k-1 \rbrace \\ \vert \Delta _{g}^ {\frac{k}{2}} \phi _{\epsilon } \vert \le \frac{c_k}{\epsilon ^k } \end{array} \right. \end{aligned}$$

where $B_{\epsilon } (p)$ is the open ball centered at p and of radius $\epsilon$ and $c_l$ are constants that do not depend on $\epsilon$. In order to get such a family, we might fix some $\phi _{\epsilon _0}$ as above, for instance, radially symmetric, and set then, for $\epsilon \le \epsilon _0, \quad \phi _\epsilon =\phi _{\epsilon _0}(\frac{r}{\epsilon })$ where r is the distance on $\mathbb {S}^n$ from $x_0$ to x. Let $u_\epsilon = \phi _\epsilon u_0$, one easily gets

$$\begin{aligned} \underset{\epsilon \longrightarrow 0}{\lim }\frac{\int \nolimits _{\mathbb {S}^n}u_\epsilon P_{{h}}(u_\epsilon )dv_{{h}}}{(\int \nolimits _{\mathbb {S}^n}{\vert u_\epsilon \vert }^{N}dv_{{h}})^{\frac{2}{N}}} =\frac{\int \nolimits _{\mathbb {S}^n}u_0 P_{{h}}(u_0)dv_{{h}}}{(\int \nolimits _{\mathbb {S}^n}{\vert u_0 \vert }^{N}dv_{{h}})^{\frac{2}{N}}} . \end{aligned}$$

As in the proof of Proposition (3.1), one has

$$\begin{aligned} \underset{\epsilon \longrightarrow 0}{\lim }\frac{1}{ \epsilon ^{2k} } V_h(B_{2\epsilon }(p)\backslash B_\epsilon (p)) =0 \quad \text {since} \quad n > 2k, \end{aligned}$$

where $V(\Omega )$ stands for the volume of $\Omega$ with respect to h. Choosing $\epsilon$ sufficiently small, it follows from (49) and (50) that there exists $\tilde{u}_\epsilon \in D_{k}^{2}(\mathbb {R}^{n})$ of the form

$$\begin{aligned} \tilde{u}_\epsilon = (u_\epsilon \circ \phi ^{-1})\varphi , \end{aligned}$$

such that

$$\begin{aligned} \frac{\int \nolimits _{\mathbb {R}^{n}}\vert \Delta ^{\frac{k}{2}} \tilde{u}_\epsilon \vert ^2dx}{(\int \nolimits _{\mathbb {R}^{n}}{\vert \tilde{u}_\epsilon \vert }^{N}dx)^{\frac{2}{N}}} < {K_{0}}^{-1} \end{aligned}$$

and this contradicts (20) see the definition (2.4). Consequently,

$$\begin{aligned} \underset{u \in C^{\infty }(\mathbb {S}^n),u \ne 0}{\inf }\frac{\int \nolimits _{\mathbb {S}^n}uP_{{h}}(u)dv_{{h}}}{(\int \nolimits _{S^{n}}{\vert u \vert }^{N}dv_{{h}})^{\frac{2}{N}}} = {K_{0}}^{-1}. \end{aligned}$$

$\square$

In the following proposition, we are going to show that there are certain manifolds such that the assumption $1 - \mu K_{0} > 0$ holds.

Proposition 6.2

Let $(\mathbb {S}^n, h)$ be the standard unit n-sphere of $\mathbb {R}^{n+1}$ and let $G_p$ be the subgroup of $O(n+1)$ of $\mathbb {R}^{n+1}$. Let $\mathbb {M}_p=\mathbb {S}^n / G_p$ be the quotient manifold and $h_p$ is the quotient metric on $\mathbb {M}_p$. Then the GJMS invariant of $\mathbb {M}_p$ satisfy the following inequality

$$\begin{aligned} \mu (\mathbb {M}_p,h_p) < {K_{0}}^{-1}. \end{aligned}$$

(51)

Proof

Writing $n=2m+1$, we let $\lbrace z_j \rbrace$, be the natural complex coordinates on $\mathbb {C}$ where $j = 1,\ldots , m+1$. Given $p\ge 2$ integer and let $G_p$ be the subgroup of $O(n+1)$ generated by

$$\begin{aligned} z_j\longrightarrow e^{\frac{2 \pi i }{p}}. \end{aligned}$$

It is easily seen that $G_p$ acts freely on $\mathbb {S}^n$. We let $\mathbb {M}_p=\mathbb {S}^n / G_p$ be the quotient manifold. We let $u_p = \overline{u} / G_p$ be the quotient function induced by $\overline{u}$ on $\mathbb {M}_p$ where $\overline{u}$ is a smooth function on $\mathbb {S}^n$. Noting that

$$\begin{aligned} \int _{\mathbb {M}_p} \vert T{u_p} \vert ^s = \dfrac{1}{p} \int _{\mathbb {S}^n} \vert T\overline{u} \vert ^s \end{aligned}$$

(52)

where s is any real number, and T is either the identity operator, the gradient operator, or the Laplace-beltrami operator. From (52) and for any $p \ge 2$, one gets

$$\begin{aligned} \mu (\mathbb {M}_p,h_p)= & {} \underset{u_p\in C^{\infty }(\mathbb {M}_p),u_p\ne 0}{\inf }\frac{\int \nolimits _{\mathbb {M}_p}u_pP_{h_p}(u_p)dv_{h_p}}{(\int \nolimits _{\mathbb {M}_p} |u_p |^{N}dv_{h_p})^{\frac{2}{N}}}\\= & {} \frac{\frac{1}{p}}{(\frac{1}{p})^{\frac{2}{N}}} \quad \underset{\overline{u}\in C^{\infty }(\mathbb {S}^n),\overline{u}\ne 0}{\inf }\frac{\int \nolimits _{\mathbb {S}^n}\overline{u}P_{h}(\overline{u})dv_{h}}{(\int \nolimits _{\mathbb {S}^n}|{\overline{u}}|^{N}dv_{h})^{\frac{2}{N}}}\\\le & {} \frac{1}{2^{\frac{2k}{n}}} \quad \underset{\overline{u}\in C^{\infty }(\mathbb {S}^n),\overline{u}\ne 0}{\inf }\frac{\int \nolimits _{\mathbb {S}^n}\overline{u}P_{h}(\overline{u})dv_{h}}{(\int \nolimits _{\mathbb {S}^n}|{\overline{u}}|^{N}dv_{h})^{\frac{2}{N}}}\\< & {} \underset{\overline{u}\in C^{\infty }(\mathbb {S}^n),\overline{u}\ne 0}{\inf }\frac{\int \nolimits _{\mathbb {S}^n}\overline{u}P_{h}(\overline{u})dv_{h}}{(\int \nolimits _{\mathbb {S}^n}|{\overline{u}}|^{N}dv_{h})^{\frac{2}{N}}} \quad \text {since} \quad n>2k. \end{aligned}$$

By Proposition(6.1), we get that $\mu (\mathbb {M}_p,h_p) < {K_{0}}^{-1}$. This ends the proof of the proposition. $\square$

It is natural to conjecture that one has the following inequality $\mu < {K_{0}}^{-1}$ for all compact Riemannian manifold but at our knowledge, this problem is still open and seems to be hard. However, we think that is very easy to prove the large inequality $\mu \le {K_{0}}^{-1}$ by following Aubin’s strategy and we have equality in this inequality if and only if (M, g) is the standard unit n-sphere $\mathbb {S}^n$ of $\mathbb {R}^{n+1}$ equipped with its round metric.