\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 63, pp. 1--23.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/63\hfil Existence of solutions] {Existence of solutions to singular fourth-order elliptic equations} \author[M. Benalili, K. Tahri \hfil EJDE-2013/63\hfilneg] {Mohammed Benalili, Kamel Tahri} % in alphabetical order \address{Mohammed Benalili Faculty of Sciences, Mathematics Dept., University Aboubakr BelKa\"{\i}d, Tlemcen, Algeria} \email{m\_benalili@mail.univ-tlemcen.dz} \address{Kamel Tahri Faculty of Sciences, Mathematics Dept., University Aboubakr Belka\"{\i}d, Tlemcen, Algeria} \email{tahri\_kamel@yahoo.fr} \thanks{Submitted October 3, 2012. Published March 1, 2013.} \subjclass[2000]{58J05} \keywords{Fourth-order elliptic equation; Hardy-Sobolev inequality; \hfill\break\indent critical Sobolev exponent} \begin{abstract} Using a method developed by Ambrosetti et al \cite{1,2} we prove the existence of weak non trivial solutions to fourth-order elliptic equations with singularities and with critical Sobolev growth. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Fourth-order elliptic equations have been widely studied, because of their importance in the analysis on manifolds particularly those involving the Paneitz-Branson operators; see for example \cite{1,2,3,4,5,6,7,8,9,10,13,16}. Different techniques have been used for solving fourth-order equations, as example the variational method which was developed by Yamabe to solve the problem of the prescribed scalar curvature. Let $(M,g)$ a compact smooth Riemannian manifold of dimension $n\geq 5$ with a metric $g$. We denote by $H_2^2(M)$ the standard Sobolev space which is the completed of the space $C^{\infty }( M)$ with respect to the norm $\| \varphi \| _{2,2}=\sum_{k=0}^{k=2}\|\nabla ^{k}\varphi \| _2.$ $H_2^2(M)$ will be endowed with the suitable equivalent norm $\| u\| _{H_2^2(M)}=\Big(\int_{M}( ( \Delta _{g}u) ^2+| \nabla _{g}u| ^2+u^2)dv_{g}\Big)^{1/2}.$ In 1979, Vaugon \cite{17} proved the existence of a positive value $\lambda$ and a non trivial solution $u\in C^{4}( M)$ to the equation $\Delta _{g}^2u-\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=\lambda f(t,x)$ where $a$, $b$ are smooth functions on $M$ and $f(t,x)$ is odd and increasing function in $t$ fulfilling the inequality $| f(t,x)| 0 in some interval. In 2001, Caraffa \cite{12} obtained the existence of a non trivial solution of class C^{4,\alpha }, \alpha \in (0,1)  for the equation \[ \Delta _{g}^2u-\nabla ^{\alpha }( a(x)\nabla _{\alpha }u) +b(x)u=\lambda f(x)| u| ^{N-2}u$ with $\lambda >0$, first for $f$ a constant and next for a positive function $f$ on $M$. Recently the first author \cite{4} showed the existence of at least two distinct non trivial solutions in the subcritical case and a non trivial solution in the critical case for the equation $\Delta _{g}^2u-\nabla ^{\alpha }( a(x)\nabla _{\alpha }u) +b(x)u=f(x)| u| ^{N-2}u$ where $f$ is a changing sign smooth function and $a$ and $b$ are smooth functions. In \cite{6} the same author proved the existence of at least two non trivial solutions to $\Delta _{g}^2u-\nabla ^{\alpha }( a(x)\nabla _{\alpha }u) +b(x)u=f(x)| u| ^{N-2}u+| u| ^{q-2}u+\varepsilon g(x)$ where $a$, $b$, $f$, $g$ are smooth functions on $M$ with $f>0$, $20$ and $\epsilon >0$ small enough. Let $S_{g}$ denote the scalar curvature of $M$. In 2011, the authors proved the following result \begin{theorem}[\cite{8}]\label{thm1} Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 6$ and $a$, $b$, $f$ smooth functions on $M$, $\lambda \in ( 0,\lambda _{\ast })$ for some specified $\lambda _{\ast}>0$, $10$ on $M$. \item[(2)] At the point $x_0$ where $f$ attains its maximum, we suppose that for $n=6$, $S_{g}(x_0)+3a(x_0)>0$, and for $n>6$ $\Big( \frac{( n^2+4n-20) }{2(n+2)(n-6)}S_{g}(x_0)+\frac{ (n-1)}{(n+2)(n-6)}a(x_0)-\frac{1}{8}\frac{\Delta f(x_0)}{ f(x_0)}\Big) >0.$ \end{itemize} Then the equation $\Delta _{g}^2u+\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=\lambda | u| ^{q-2}u+f(x)| u| ^{N-2}u$ admits a non trivial solution of class $C^{4,\alpha }( M)$, $\alpha \in (0,1)$. \end{theorem} Recently Madani \cite{14} studied the Yamabe problem with singularities when the metric $g$ admits a finite number of points with singularities and is smooth outside these points. More precisely, let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$, we denote by $T^{\ast }M$ the cotangent space of $M$. The space $H_2^{p}(M,T^{\ast}M\otimes T^{\ast }M)$ is the set of sections $s$ ($2$-covariant tensors) such that in normal coordinates the components $s_{ij}$ of $s$ are in $H_2^{p}$ the complement of the space $C_0^{\infty }(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|_{2,p}=\sum_{k=0}^{k=2}\| \nabla ^{k}\varphi \| _p$. Solving the singular Yamabe problem is equivalent to finding a positive solution $u\in H_2^{p}(M)$ of the equation $$\Delta _{g}u+\frac{n-2}{4(n-1)}S_{g}u=k|u|^{N-2}u, \label{*}$$ where $S_{g}$ is the scalar curvature of the $g$ and $k$ is a real constant. The Christoffels symbols belong to $\ H_{1}^{p}( M)$, the Riemannian curvature tensor, the Ricci tensor $Ric_{g}$ and scalar curvature $S_{g}$ are in $L^{p}(M)$, hence equation \ref{*} is the singular Yamabe equation. Under the assumptions that $g$ is a metric in the Sobolev space $H_2^{p}(M,T^{\ast}M\otimes T^{\ast }M)$ with $p>n/2$ and that there exist a point $P\in M$ and $\delta >0$ such that $g$ is smooth in the ball $B_p(\delta )$, Madani \cite{14} proved the existence of a metric $\overline{g}=u^{N-2}g$ conformal to $g$ such that $u\in H_2^{p}(M)$, $u>0$ and the scalar curvature $S_{\overline{g}}$ of $\overline{g}$ is constant if $(M,g)$ is not conformal to the round sphere. The author in \cite{7} considered fourth-order elliptic equations, with singularities, of the form $$\Delta ^2u-\nabla ^{i}( a(x)\nabla _{i}u) +b(x)u=f|u| ^{N-2}u \label{1}$$ where the functions $a$ and $b$ are in $L^{s}(M)$, $s>\frac{n}{2}$ and in $L^{p}(M)$, $p>\frac{n}{4}$ respectively, $N=\frac{2n}{n-4}$ is the Sobolev critical exponent in the embedding $H_2^2( M) \hookrightarrow L^{N}( M)$. He established the following results. Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold, $n\geq 6$, $a\in L^{s}(M)$, $b\in L^{p}(M)$, with $s>\frac{n}{2}$, $p>\frac{n}{4}$, $f\in C^{\infty }(M)$ a positive function and $x_0\in M$ such that $f(x_0)=\max_{x\in M}f(x)$. \begin{theorem} For $n\geq 10$, or $n=8,9$ and $20. \] For$n=6$and$\frac{3}{2}0. \] Then \eqref{1} has a non trivial weak solution $u$ in $H_2^2( M)$. Moreover if $a\in H_{1}^{s}( M)$, then $u\in$ $C^{0,\beta }( M)$, for some $\beta \in ( 0,1- \frac{n}{4p})$. \end{theorem} In this article, we extend results obtained in Theorem \ref{thm1} to the case of singular elliptic fourth order, more precisely we are concerned with the following problem: Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$. Let $a\in L^{r}(M)$, $b\in L^{s}(M)$ where $r>\frac{n}{2}$, $s>\frac{n}{4}$ and $f$ a positive $C^{\infty }$-function on $M$; we look for non trivial solution of the equation $$\Delta _{g}^2u+\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=\lambda | u| ^{q-2}u+f(x)| u| ^{N-2}u \label{2}$$ where $10$ a real number. In case the $\lambda =0$ and $a=\frac{4}{n-2}R_{ic_{g}}-\frac{( n-2) ^2+4}{2( n-1)( n-2) }S_{g}.g, \quad b=\frac{n-4}{2}Q_{g}^{n},$ where $Q_{g}^{n}=\frac{1}{2( n-1) }\Delta S_{g}+\frac{n^{3}-4n^2+16n-16 }{8( n-1) ^2( n-2) ^2}S_{g}^2-\frac{2}{( n-2) ^2}| Ric_{g}| ^2$ suppose that $g$\ is a metric in the Sobolev space $H_{4}^{p}(M,T^{\ast}M\otimes T^{\ast }M)$\ with $\ p>\frac{n}{4}$, then the Ricci $Ric_{g}$ curvature and the scalar curvature $S_{g}$ are in the Sobolev spaces $H_2^{p}(M,T^{\ast }M\otimes T^{\ast }M)$ and $H_2^{p}(M)$ respectively, hence $b\in L^{s}( M)$ with $s>\frac{n}{4}$ and by Sobolev embedding $a\in L^{r}( M)$ with $r>\frac{n}{2}$. In this latter case the equation $$\Delta _{g}^2u+\operatorname{div}_{g}( a(x)\nabla _{g}u) +b(x)u=f(x)| u| ^{N-2}u \label{**}$$ is called singular $Q$-curvature equation$.$For more general coefficients $a\in L^{r}( M)$ with $r>\frac{n}{2}$ and $b\in L^{s}(M)$ with $s>\frac{n}{4}$, the equation \eqref{**} is called singular $Q$-curvature type equation. To solve equation \eqref{2}, we use a method developed in \cite{1} and \cite{2} which resumes to study the variations of functional associated to equation \ref{2} on the manifold $M_{\lambda }$ defined in section 2. Serious difficulties appear compared with the smooth case: considering the equation \eqref{12} in section 4, we need a Hardy-Sobolev inequality and Releich-Kondrakov embedding on a manifolds. In the case of the singular Yamabe equation theses latters were established in \cite{14} and in the case of singular $Q$-curvature type equations by the first author in \cite{7}. In the sharp cases (see section 5) the Hardy Sobolev inequality and the Releich-Kondrakov embedding are no more valid so we need an additional assumption with some tricks combined with the Lebesgue dominated convergence theorem. Denote by $P_{g}$ the operator defined in the weak sense on $H_2^2(M)$ by $P_{g}(u)=\Delta ^2u+\operatorname{div}(a\nabla u)+bu$. $P_{g}$ is called coercive if there exits $\Lambda >0$ such that for any $u\in H_2^2(M)$ $\int_{M}uP_{g}(u)dv_{g}\geq \Lambda \| u\|_{H_2^2(M)}^2.$ Our main result reads as follows. \begin{theorem} Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 6$ and $f$ a positive function. Suppose that $P_{g}$ is coercive and at a point $x_0$ where $f$ attains its maximum the following two conditions hold: \begin{gathered} \begin{aligned} \frac{\Delta f(x_0)}{f( x_0) } &<\Big(\frac{n(n^2+4n-20) }{3( n+2) ( n-4) ( n-6) } \frac{1}{( 1+\| a\| _r+\| b\|_s) ^{n/4}}\\ &\quad -\frac{n-2}{3( n-1) }\Big) S_{g}( x_0) \quad \text{when }n>6, \end{aligned} \\ S_{g}(x_0)>0\quad \text{when }n=6. \end{gathered} \label{C} Then there is $\lambda ^{\ast }>0$ such that for any $\lambda \in (0, \lambda ^{\ast })$, the equation \eqref{2} has a non trivial weak solution. \end{theorem} For fixed $R\in M$, we define the function $\rho$ on $M$ by $$\rho (Q)=\begin{cases} d(R,Q)&\text{if } d(R,Q)<\delta (M) \\ \delta (M)&\text{if }d(R,Q)\geq \delta (M) \end{cases} \label{3}$$ where $\delta (M)$ denotes the injectivity radius of $M$. For real numbers $\sigma$ and $\mu$, consider the following equation, in the distribution sense, $$\Delta ^2u-\nabla ^{i}(\frac{a}{\rho ^{\sigma }}\nabla _{i}u)+\frac{bu}{ \rho ^{\mu }}=\lambda | u| ^{q-2}u+f(x)| u| ^{N-2}u \label{4}$$ where the functions $a$ and $b$ are smooth on $M$. \begin{corollary} \label{coro1} Let $0<\sigma <\frac{n}{r}<2$ and $0<\mu <\frac{n}{s}<4$. Suppose that \begin{gather*} \frac{\Delta f(x_0)}{f( x_0) }<\frac{1}{3}\Big( \frac{ (n-1)n( n^2+4n-20) }{( n^2-4) ( n-4) ( n-6) }\frac{1}{( 1+\| a\| _r+\| b\| _s) ^{n/4}}-1\Big) S_{g}( x_0) \\ \text{when } n>6 , \\ S_{g}(x_0)>0 \quad \text{when }n=6. \end{gather*} Then there is $\lambda _{\ast }>0$ such that if $\lambda \in (0,\lambda _{\ast })$, the \eqref{4} possesses a weak non trivial solution $u_{\sigma ,\mu }\in M_{\lambda }$. \end{corollary} In the sharp case $\sigma =2$ and $\mu =4$, letting $K(n,2,\gamma )$ be the best constant in the Hardy-Sobolev inequality given by Theorem \ref{thm7} we obtain the following result. \begin{theorem} Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$. Let $( u_{_{\sigma _{m},\mu _{m}}}) _{m}$ be a sequence in $M_{\lambda}$ such that \begin{gather*} J_{\lambda ,\sigma ,\mu }(u_{_{\sigma _{m},\mu _{m}}})\leq c_{\sigma ,\mu } \\ \nabla J_{\lambda }(u_{_{\sigma ,\mu }})-\mu _{_{\sigma ,\mu }}\nabla \Phi _{\lambda }(u_{_{\sigma ,\mu }})\to 0 \end{gather*} Suppose that $c_{\sigma ,\mu }<\frac{2}{nK_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}$ and $1+a^{-}\max ( K(n,2,\sigma ),A( \varepsilon ,\sigma ) ) +b^{-}\max ( K(n,2,\mu ),A( \varepsilon ,\mu )) >0$ then the equation $\Delta ^2u-\nabla ^{\mu }(\frac{a}{\rho ^2}\nabla _{\mu }u)+\frac{bu}{ \rho ^{4}}=f| u| ^{N-2}u+\lambda | u| ^{q-2}u$ in the distribution has a weak non trivial solution. \end{theorem} Our paper is organized as follows: in a first section we show that the manifold of constraints is non empty, in the second one we establish a generic existence result to equation \ref{2}. The third section deals with applications to particular equations which could arise from conformal geometry. In the fourth section and under supplementary assumption we obtain non trivial solution in the critical case. The last section is devoted to tests functions which verify geometric assumptions and by the same way complete the proofs of our claimed theorems in the introduction. \section{The manifold $M_{\protect\lambda }$ of constraints is non empty} In this section, we consider on $H_2^2(M)$ the functional $J_{\lambda }(u)=\frac{1}{2}\int_{M}( | \Delta _{g}u| ^2-a(x)| \nabla _{g}u| ^2+b(x)u^2) dv_{g}- \frac{\lambda }{q}\int_{M}| u| ^qdv_{g}-\frac{1}{N} \int_{M}f(x)| u| ^{N}dv_{g}$ associated to Equation \ref{2}. First, we put $\Phi _{\lambda }(u)=\langle \nabla J_{\lambda }(u),\text{ } u\rangle$ hence $\Phi _{\lambda }(u)=\int_{M}( ( \Delta _{g}u) ^2-a(x)| \nabla _{g}u| ^2+b(x)u^2) dv_{g}-\lambda \int_{M}| u| ^qdv_{g}-\int_{M}f(x)| u| ^{N}dv_{g}.$ We let $M_{\lambda }=\{ u\in H_2^2(M):\text{ }\Phi _{\lambda }(u)=0\text{ and }\| u\| \geq \tau >0\} .$ \begin{proposition} \label{prop1} The norm $$\| u\| =(\int_{M}| \Delta _{g}u| ^2-a(x)| \nabla _{g}u| ^2+b(x)u^2dv_{g})^{1/2}$$ is equivalent to the usual norm on $H_2^2(M)$ if and only if $P_{g}$ is coercive. \end{proposition} \begin{proof} If $P_{g}$ is coercive there is $\Lambda >0$ such that for any $u\in H_2^2(M)$, $\int_{M}P_{g}(u)udv_{g}\geq \Lambda \| u\|_{H_2^2(M)}^2$ and since $a\in L^{r}(M)$ and $b\in L^{s}(M)$ where $r>\frac{n}{2}$ and $s>\frac{n}{4}$, by H\"{o}lder's inequality we obtain $\int_{M}uP_{g}(u)dv_{g}\leq \| \Delta _{g}u\| _2^2+\| a\| _{\frac{n}{2}}\| \nabla _{g}u\| _{2^{\ast }}^2+\| b\| _{\frac{n}{4} }\| u\| _{N}^2$ where $2^{\ast }=2n/(n-2)$. The Sobolev's inequalities lead to: for any $\eta >0$, $\| \nabla _{g}u\| _{2^{\ast }}^2\leq \max ((1+\eta )K(n,1)^2,A_{\eta })\int_{M}( | \nabla _{g}^2u| ^2+| \nabla _{g}u| ^2) dv_{g}$ where $K(n,1)$ denotes the best Sobolev's constant in the embedding $H_{1}^2( \mathbb{R}^n) \hookrightarrow L^{\frac{2n}{n-2}}(\mathbb{R}^n)$, and for any $\epsilon >0$, $\| u\| _{N}^2\leq \max ((1+\varepsilon)K_0,B_{\varepsilon }) \| u\| _{H_2^2(M)}^2$ where in this latter inequality $K_0$ is the best Sobolev's constant in the embedding $H_{1}^2( M) \hookrightarrow L^{\frac{2n}{n-2} }( M)$ and $B_{\epsilon }$ the corresponding (see \cite{3}). Now by the well known formula (see \cite[page 115]{3}) $\int_{M}| \nabla _{g}^2u| ^2dv_{g}=\int_{M}(| \Delta _{g}u| ^2-R_{ij} \nabla ^{i}u\nabla ^{j}u) dv_{g}$ where $R_{ij}$ denote the components of the Ricci curvature, there is a constant $\beta >0$ such that $\int_{M}| \nabla _{g}^2u| ^2dv_{g}\leq \int_{M}| \Delta _{g}u| ^2+\beta | \nabla_{g}u| ^2dv_{g}$ so we obtain $\| \nabla _{g}u\| _{2^{\ast }}^2\leq (\beta +1)\max ((1+\eta )K(n,1)^2,A_{\eta })\int_{M}( | \Delta _{g}u| ^2+| \nabla _{g}u| ^2+u^2) dv_{g}$ and we infer that \begin{align*} \int_{M}P_{g}(u)udv_{g} &\leq \| u\|_{H_2^2(M)}^2+(\beta +1)\| a\| _{\frac{n}{2}} \max((1+\eta )K(n,1)^2,A_{\eta })\| u\| _{H_2^2(M)}^2\\ &\quad +\| b\| _{\frac{n}{4}}\max ((1+\varepsilon )K_0,B_{\varepsilon })\| u\| _{H_2^2(M)}^2. \end{align*} Hence \begin{align*} &\int_{M}uP_{g}(u)dv_{g}\\ &\leq \underbrace{\max ( 1,\| b\| _{\frac{n}{4}}\max ((1+\varepsilon )K_0,B_{\varepsilon }),(\beta +1)\| a\| _{\frac{n}{2}}\max ((1+\varepsilon )K(n,1)^2,A_{\varepsilon })) }_{>0}\\ &\quad\times \| u\| _{H_2^2(M)}^2. \end{align*} \end{proof} \begin{lemma}\label{lem0} The set $M_{\lambda }$ is non empty provided that $\lambda \in (0,\lambda _0)$ where $\lambda _0=\frac{( 2^{q-2}-2^{q-N}) \Lambda ^{\frac{N-q}{ N-2}}}{V(M)^{(1-\frac{q}{N})}( \max_{x\in M}f(x)) ^{\frac{2-q}{N-2 }}(\max ((1+\varepsilon )K( n,2) ,A_{\varepsilon }))^{\frac{N-q}{ N-2}}}.$ \end{lemma} \begin{proof} The proof of this lemma is the same as in \cite{8}, but we give it here for convenience. Let $t>0$ and $u\in H_2^2(M)-\{ 0\}$. Evaluating $\Phi _{\lambda }$ at $tu$, we obtain $\Phi _{\lambda }(tu)=t^2\| u\| ^2-\lambda t^q\| u\| _{q}^q-t^{N}\int_{M}f(x)| u| ^{N}dv_{g}.$ Put \begin{gather*} \alpha (t)=\| u\|^2-t^{N-2}\int_{M}f(x)| u| ^{N}dv(g),\\ \beta (t)=\lambda t^{q-2}\| u\| _{q}^q\text{;} \end{gather*} by Sobolev's inequality, we obtain $\alpha (t)\geq \| u\| ^2-\max_{x\in M}f(x)(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{N/2} \|u\| _{H_2^2(M)}^{N}t^{N-2}.$ By the coercivity of the operator $P_{g}=\Delta _{g}^2-\operatorname{div}_{g}(a\nabla _{g}) +b$ there is a constant $\Lambda >0$ such that $\alpha (t)\geq \| u\| ^2-\Lambda ^{-N/2}\max_{x\in M}f(x) (\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{ \frac{N}{2}}\| u\| ^{N}t^{N-2}.$ Letting $\alpha _{1}(t)=\| u\| ^2-\Lambda ^{-N/2}\max_{x\in M}f(x) (\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{N/2}\| u\| ^{N}t^{N-2}$ H\"{o}lder and Sobolev inequalities lead to $\beta (t)\leq \lambda V(M)^{(1-\frac{q}{N})}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\| u\| _{H_2^2(M)}^qt^{q-2}$ and the coercivity of $P_{g}$ assures the existence of a constant $\Lambda>0$ such that $\beta (t)\leq \lambda \Lambda ^{-q/2}V(M)^{(1-\frac{q}{N})}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\| u\| ^qt^{q-2}.$ Put $\beta _{1}(t)=\lambda \Lambda ^{-q/2}V(M)^{(1-\frac{q}{N})}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\| u\| ^qt^{q-2}.$ Let $t_0$ such $\alpha _{1}(t_0)=0$; i.e., $t_0=\frac{\Lambda ^{\frac{N}{2(N-2)}}}{\| u\| ( \max_{x\in M}f(x)) ^{\frac{1}{N-2}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{N}{2(N-2)}}}$ Now since $\alpha _{1}(t)$ is a decreasing and a concave function and $\beta_{1}(t)$ is a decreasing and convex function, then \begin{gather*} \min_{t\text{ }\in (0,\text{ }\frac{t_0}{2}]}\alpha _{1}(t)=\alpha _{1}(\frac{t_0}{2})=\| u\| ^2(1-2^{2-N})>0,\\ \min_{t\text{ }\in (0,\text{ }\frac{t_0}{2}]}\beta _{1}(t)=\beta _{1}( \frac{t_0}{2})>0, \end{gather*} where $\beta _{1}(\frac{t_0}{2}) =\frac{2^{2-q}\lambda V(M)^{(1-\frac{q}{N})} \Lambda ^{\frac{q-N}{N-2}}\| u\| ^2}{(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{q-N}{N-2}}( \max_{x\in M}f(x)) ^{\frac{q-2}{N-2}}}.$ Consequently $\Phi _{\lambda }(tu)=0$ with $t\in (0,$ $\frac{t_0}{2}]$ has a solution if $\min_{t\in (0,\frac{t_0}{2}]}\alpha _{1}(t)\geq \max_{t\in (0,\frac{t_0}{2}]}\beta _{1}(t);$ that is to say $0<\lambda <\frac{( 2^{q-2}-2^{q-N}) ( \max_{x\in M}f(x)) ^{\frac{q-2}{N-2}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{q-N}{N-2}}}{\Lambda ^{\frac{N-q}{N-2}}V(M)^{(1- \frac{q}{N})}}=\lambda _0$ Let $t_{1}\in (0,\frac{t_0}{2}]$ such that $\Phi _{\lambda }(t_{1}u)=0$. If we take $u\in H_2^2( M)$ such that $\| u\| \geq \frac{\rho }{t_{1}}$ and $v=t_{1}u$ we obtain $\Phi _{\lambda }(v)=0$ and $\| v\| =t_{1}\|u\| \geq \rho$; i.e., $v\in M_{\lambda }$ provided that $\lambda \in (0,\lambda _0)$. \end{proof} \section{Existence of non trivial solutions in $M_{\lambda }$} The following lemmas whose proofs are similar modulo minor modifications as in \cite{8} give the geometric conditions to the functional $J_{\lambda }$. \begin{lemma}\label{lem1} Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$. For all $u\in M_{\lambda }$ and all $\lambda \in ( 0,\min( \lambda _0,\lambda _{1}) )$ there is $A>0$ such that $J_{\lambda }(u)\geq A>0$ where $\lambda _{1}=\frac{\frac{( N-2) q}{2( N-q) }\Lambda ^{q/2}} {V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K(n,2),A_{\varepsilon }))^{q/2}\tau ^{q-2}}.$ \end{lemma} \begin{lemma}\label{lem2} Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$. The following assertions are true: \begin{itemize} \item[(i)] $\langle \nabla \Phi _{\lambda }(u),u\rangle <0$ for all $u\in M_{\lambda }$ and for all $\lambda \in (0,\min (\lambda _0,\lambda _{1}))$. \item[(ii)] The critical points of $J_{\lambda }$ are points of $M_{\lambda }$. \end{itemize} \end{lemma} Now, we show that $J_{\lambda }$ satisfies the Palais-Smale condition on $M_{\lambda }$ provided that $\lambda >0$ is sufficiently small. The result is given by the following lemma whose proof is different from the one in the case of smooth coefficients. \begin{lemma}\label{lem3} Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 5$. Let $( u_{m}) _{m}$ be a sequence in $M_{\lambda }$ such that \begin{gather*} J_{\lambda }(u_{m})\leq c \\ \nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda }(u_{m})\to 0. \end{gather*} Suppose that $c<\frac{2}{nK_0^{n/4}(f(x_0))^{(n-4)/4}}$ then there is a subsequence $( u_{m}) _{m}$ converging strongly in $H_2^2(M)$. \end{lemma} \begin{proof} Let $( u_{m}) _{m}\subset M_{\lambda }$ and $J_{\lambda }(u_{m})=\frac{N-2}{2N}\| u_{m}\| ^2-\lambda \frac{N-q}{Nq}\int_{M}| u_{m}| ^qdv_{g}\,.$ As in the proof of Lemma \ref{lem2}, we have \begin{gather*} J_{\lambda }(u_{m})\geq \frac{N-2}{2N}\| u_{m}\| ^2-\lambda \frac{N-q}{Nq}\Lambda ^{-q/2}V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\| u_{m}\| ^q, \\ \begin{aligned} J_{\lambda }(u_{m})&\geq \| u_{m}\| ^2(\frac{N-2}{2N} -\lambda \frac{N-q}{Nq}\Lambda ^{-q/2}V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\tau ^{q-2})\\ &>0\,. \end{aligned} \end{gather*} Since $0<\lambda <\frac{\frac{( N-2) q}{2( N-q) } \Lambda ^{q/2}}{V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K(n,2),A_{\varepsilon }))^{q/2}\tau ^{q-2}}$ and $J_{\lambda }(u_{m})\leq c$, we obtain \begin{align*} c&\geq J_{\lambda }(u_{m}) \\ &\geq \big[ \frac{N-2}{2N}-\lambda \frac{N-q}{Nq}\Lambda ^{-\frac{q}{2} }V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{\frac{q }{2}}\tau ^{q-2}\big]\| u_{m}\| ^2>0 \end{align*} so $\| u_{m}\| ^2\leq \frac{c}{\frac{N-2}{2N}-\lambda \frac{ N-q}{Nq}\Lambda ^{-q/2}V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K_0,A_{\varepsilon }))^{q/2}\tau ^{q-2}}<+\infty .$ Then $( u_{m}) _{m}$ is a bounded in $H_2^2(M)$. By the compactness of the embedding $H_2^2(M)\subset H_p^{k}(M)$ ($k\ =0,1$; $p0$ on $M_{\lambda }$ with $\lambda$ is as in Lemma \ref{lem1} and by hypothesis, $c\geq J_{\lambda }( u_{m}) >( J_{\lambda }( u_{m}) -J_{\lambda }( u) ) =\frac{2}{n}\int_{M}( \Delta _{g}(u_{m}-u)) ^2dv_{g}$ and $c<\ \frac{2}{n K_0^{n/4}(\max_{x\in M}f(x))^{\frac{n}{4}-1}}.$ It is obvious that $\Phi _{\lambda }(u)=0\text{ \ and }\| u\| \geq \tau$ i.e. $u\in M_{\lambda }$. \end{proof} Now we show the existence of a sequence in $M_{\lambda }$ satisfying the conditions of Palais-Smale. \begin{lemma}\label{lem4} Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 5$, then there is a couple $( u_{m},\mu _{m}) \in M_{\lambda }\times R$ such that $\nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda }(u_{m})\to 0$ strongly in $(H_2^2(M))^{\ast }$ and $J_{\lambda }( u_{m})$ is bounded provide that $\lambda \in( 0,\lambda _{\ast })$ with $\lambda _{\ast }=\{ \min (\lambda _0,\lambda _{1}),0\}$. \end{lemma} \begin{proof} Since $J_{\lambda }$ is Gateau differentiable and by Lemma \ref{lem1} bounded below on $M_{\lambda }$ it follows from Ekeland's principle \ that there is a couple $(u_{m},$ $\mu _{m})$ $\in M_{\lambda }\times R$ such that $\nabla J_{\lambda }(u_{m})-\mu _{m}\nabla \Phi _{\lambda }(u_{m})\to 0$ strongly in $(H_2^2(M))^{^{\prime }}$ and $J_{\lambda }( u_{m})$ is bounded i.e. $(u_{m},$ $\mu _{m})_{m}$ is a Palais-Smale sequence on $M_{\lambda }$. \end{proof} Now we are in position to establish the following generic existence result. \begin{theorem}\label{thm3} Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 5$ and $f$ a positive function. Suppose that $P_{g}$ is coercive and $$c<\frac{2}{nK_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}. \label{C1}$$ Then there is $\lambda ^{\ast }>0$ such that for any $\lambda \in (0,\lambda ^{\ast })$, the equation \eqref{2} has a non trivial weak solution. \end{theorem} \begin{proof} By Lemma \ref{lem3} and \ref{lem4} there is $u\in H_2^2( M)$ such that $J_{\lambda }(u)=\min_{\varphi \in M_{\lambda }}J_{\lambda }(\varphi ).$ By Lagrange multiplicative theorem there is a real number $\mu$ such that for any $\varphi \in H_2^2( M)$, $$\langle \nabla J_{\lambda }(u),\varphi \rangle =\mu \langle \nabla \Phi _{\lambda }(u),\varphi \rangle \label{9}$$ and letting $\varphi =u$ in the equation \eqref{9}, we obtain $\Phi _{\lambda }(u)=\langle \nabla J_{\lambda }(u),u\rangle =\mu \langle \nabla \Phi _{\lambda }(u),u\rangle .$ By Lemma \ref{lem2} we obtain that $\mu =0$ and by equation \eqref{9}, we infer that for any $\varphi \in H_2^2( M)$ $\langle \nabla J_{\lambda }(u),\varphi \rangle =0$ hence $u$ is weak non trivial solution to equation \eqref{2} and since by Lemma \ref{lem2}, $u$ is a critical points of $J_{\lambda }$. We conclude that $u\in M_{\lambda }$. \end{proof} \section{Applications} Let $P\in M$, we define a function on $M$ by $$\rho _{_{P}}(Q)=\begin{cases} d(P,Q)&\text{if }d(P,Q)<\delta (M) \\ \delta (M)&\text{if }d(P,Q)\geq \delta (M) \end{cases} \label{10}$$ where $\delta (M)$ is the injectivity radius of $M$. For brevity we denote this function by $\rho$. The weighted $L^{p}( M,\rho ^{\gamma })$ space will be the set of measurable functions $u$ on $M$ such that $\rho ^{\gamma }| u| ^{p}$ are integrable where $p\geq 1$. We endow $L^{p}( M,\rho ^{\gamma })$ with the norm $\| u\| _{p,\rho }=( \int_{M}\rho ^{\gamma }| u| ^{p}dv_{g}) ^{1/p}.$ In this section we need the Hardy-Sobolev inequality and the Releich-Kondrakov embedding whose proofs are given in \cite{7}. \begin{theorem} \label{thm7} Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$ and $p$, $q$ , $\gamma$ are real numbers such that $\frac{\gamma }{p}=\frac{n}{q}-\frac{n}{p}-2$ $\$and $2\leq p\leq \frac{2n}{n-4}$. For any $\epsilon >0$, there is $A(\epsilon ,q,\gamma )$ such that for any $u\in H_2^2(M)$, $$\| u\| _{p,\rho ^{\gamma }}^2\leq (1+\epsilon )K(n,2,\gamma )^2\| \Delta _{g}u\| _2^2+A(\epsilon ,q,\gamma )\| u\| _2^2 \label{11}$$ where $K(n,2,\gamma )$ is the optimal constant. \end{theorem} In the case $\gamma =0$, $K(n,2,0)=K(n,2)=K_0^{1/2}$ is the best constant in the Sobolev's embedding of $H_2^2(M)$ in $L^{N}(M)$ where $N=\frac{2n}{n-4}$. \begin{theorem} Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 5$ and $p$, $q$, $\gamma$ are real numbers satisfying $1\leq q\leq p\leq \frac{nq}{n-2q}$, $\gamma <0$ and $l=1, 2$. If $\frac{\gamma }{p}=n$ $(\frac{1}{q}-\frac{1}{p})-l$ then the inclusion $H_{l}^q(M)\subset$ $L^{p}(M,\rho ^{\gamma })$ is continuous. If $\frac{\gamma }{p}>n$ $(\frac{1}{q}-\frac{1}{p})-l$ then inclusion $H_{l}^q(M)\subset$ $L^{p}(M,\rho ^{\gamma })$ is compact. \end{theorem} We consider the equation $$\Delta _{g}^2u+\operatorname{div}_{g}\Big( \frac{a(x)}{\rho ^{\sigma }}\nabla _{g}u\Big) +\frac{b(x)}{\rho ^{\mu }}u=\lambda | u| ^{q-2}u+f(x)| u| ^{N-2}u \label{12}$$ where $a$ and $b$ are smooth functions and $\rho$ denotes the distance function defined by \eqref{10}, $\lambda >0$ in some interval $(0,\lambda _{\ast })$, $10$ such that if $\lambda \in (0,\lambda_{\ast })$, equation \eqref{12} possesses a weak non trivial solution $u_{\sigma ,\mu }\in M_{\lambda }$. \end{theorem} \begin{proof} Let $\tilde{a}=\frac{a(x)}{\rho ^{\sigma }}$ and $\tilde{b}=\frac{b(x)}{\rho ^{\mu }}$, so if $\sigma \in (0,\min ( 2,\frac{n}{s}) )$ and $\mu \in (0,\min (4,\frac{n}{p}))$, obliviously $\tilde{a}\in L^{s}(M)$, $\tilde{b}\in L^{p}(M)$, where $s>\frac{n}{2}$ and $p>\frac{n}{4}$. Theorem \ref{thm4} is a consequence of Theorem \ref{thm3}. \end{proof} \section{The critical cases $\sigma =2$ and $\mu =4$} In the cases $\sigma =2$ and $\mu =4$ the Hardy-Sobolev inequality proved in case of manifolds by the first author in \cite{7} and is formulated in Theorem \ref{thm7} is no longer valid, so we consider the subcritical cases $0<\sigma <2$ and $0<\mu <4$ and we tend $\sigma$ to $2$and $\mu$ to $4$. This can be done successfully by adding an appropriate assumption and by using the Lebesgue dominated converging theorem. By section four, for any $\sigma \in (0,\min ( 2,\frac{n}{s}) )$ and $\mu \in (0,\min (4,\frac{n}{p}))$, there is a solution $u_{\sigma ,\mu}\in M_{\lambda }$ of equation \eqref{2}. Now we are going to show that the sequence $( u_{\sigma ,\mu }) _{\sigma ,\mu }$ is bounded in $H_2^2( M)$. Evaluating $J_{\lambda ,\sigma ,\mu }$ at $u_{\sigma ,\mu }$ $J_{\lambda ,\sigma ,\mu }(u_{\sigma ,\mu })=\frac{1}{2}\| u_{\sigma ,\mu }\| ^2-\frac{1}{N}\int_{M}f(x)| u_{\sigma ,\mu }| ^{N}dv_{g}-\frac{1}{q}\lambda \int_{M}| u_{\sigma ,\mu }| ^qdv_{g}$ and taking account of $u_{\sigma ,\mu }\in M_{\lambda }$, we infer that $J_{\lambda ,\sigma ,\mu }(u_{\sigma ,\mu })=\frac{N-2}{2N}\| u_{\sigma ,\mu }\| ^2-\lambda \frac{N-q}{Nq}\int_{M}| u_{\sigma ,\mu }| ^qdv_{g}.$ For a smooth function $a$ on $M$, denotes by $a^{-}=\min ( 0,\min_{x\in M}(a(x))$. Let $K(n,2,\sigma )$ the best constant and $A( \varepsilon ,\sigma )$ the corresponding constant in the Hardy- Sobolev inequality given in Theorem \ref{thm7}. \begin{theorem} Let $(M,g)$ be a Riemannian compact manifold of dimension $n\geq 5$. Let $( u_{m}) _{m}=( u_{\sigma _{m},\mu _{m}}) _{m}$ be a sequence in $M_{\lambda }$ such that \begin{gather*} J_{\lambda ,\sigma ,\mu }(u_{m})\leq c_{\sigma ,\mu } \\ \nabla J_{\lambda }(u_{m})-\mu _{_{\sigma ,\mu }}\nabla \Phi _{\lambda }(u_{m})\to 0. \end{gather*} Suppose that $c_{\sigma ,\mu }<\frac{2}{n\text{ }K( n,2) ^{n/4} (\max_{x\in M}f(x))^{(n-4)/4}}$ and $1+a^{-}\max ( K(n,2,\sigma ),A( \varepsilon ,\sigma ) ) +b^{-}\max ( K(n,2,\mu ),A( \varepsilon ,\mu )) >0.$ Then the equation $\Delta ^2u-\nabla ^{\mu }(\frac{a}{\rho ^2}\nabla _{\mu }u)+\frac{bu}{ \rho ^{4}}=f| u| ^{N-2}u+\lambda | u| ^{q-2}u$ has a non trivial solution in the sense of distributions. \end{theorem} \begin{proof} Let $( u_{m}) _{m}\subset M_{\lambda ,\sigma ,\mu }$, $J_{\lambda ,\sigma ,\mu }(u_{m})=\frac{N-2}{2N}\| u_{m}\| ^2-\lambda \frac{N-q}{Nq}\int_{M}| u_{m}| ^qdv_{g}$ As in proof of Theorem \ref{thm3}, we obtain \begin{align*} J_{\lambda ,\sigma ,\mu }(u_{m}) &\geq \| u_{m}\| ^2\Big(\frac{N-2}{2N}\\ &\quad -\lambda \frac{N-q}{Nq}\Lambda _{\sigma ,\mu }^{-q/2}V(M)^{1- \frac{q}{N}}(\max ((1+\varepsilon )K( n,2) , A_{\varepsilon }))^{q/2}\tau ^{q-2}\Big)>0 \end{align*} where $0<\lambda <\frac{\frac{( N-2) q}{2( N-q) } \Lambda _{\sigma ,\mu }^{q/2}}{V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K( n,2) ,A_{\varepsilon }))^{q/2}\tau ^{q-2}}.$ First we claim that $\lim_{( \sigma ,\mu ) \to ( 2^{-},4^{-}) }\inf \Lambda _{\sigma ,\mu }>0.$ Indeed, if $\nu _{1,\sigma ,\mu }$ denotes the first nonzero eigenvalue of the operator $$P_{g}=\Delta _{g}^2-\operatorname{div}( \frac{a}{\rho ^{\sigma }} \nabla _{g}) +\frac{b}{\rho ^{\mu }},$$ then clearly $\Lambda _{\sigma ,\mu }\geq \nu _{1,\sigma ,\mu }$. Suppose on the contrary that $\lim_{(\sigma ,\mu ) \to ( 2^{-},4^{-}) }\inf \Lambda _{\sigma ,\mu }=0$, then $\lim \inf_{( \sigma ,\mu ) \to ( 2^{-},4^{-}) }\nu _{1,\sigma ,\mu }=0$. Independently, if $u_{\sigma ,\mu }$ is the corresponding eigenfunction to $\nu _{1,\sigma ,\mu}$ we have \begin{aligned} \nu _{1,\sigma ,\mu } &=\| \Delta u_{\sigma,\mu}\| _2^2+\int_{M} \frac{ a| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }}dv_{g} +\int_{M}\frac{bu_{\sigma,\mu}^2}{\rho ^{\mu }}dv_{g} \\ &\geq \| \Delta u_{\sigma,\mu}\| _2^2+a^{-}\int \frac{| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }}dv_{g}+b^{-}\int_{M} \frac{u_{\sigma,\mu}^2}{\rho ^{\mu }}dv_{g} \end{aligned} \label{13} where $a^{-}=\min ( 0,\min_{x\in M}a(x))$ and $b^{-}=\min (0,\min_{x\in M}b(x))$. The Hardy- Sobolev's inequality given by Theorem \ref{thm7} leads to $\int_{M}\frac{| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }} dv_{g}\leq C(\| \nabla | \nabla u_{\sigma,\mu}| \| ^2+\| \nabla u_{\sigma,\mu}\| ^2),$ and since $\| \nabla | \nabla u_{\sigma,\mu}| \| ^2\leq \| \nabla ^2u_{\sigma,\mu}\| ^2\leq \| \Delta u_{\sigma,\mu}\| ^2+\beta \| \nabla u_{\sigma,\mu}\| ^2$ where $\beta >0$ is a constant and it is well known that for any $\varepsilon >0$ there is a constant $c( \varepsilon ) >0$ such that $\| \nabla u_{\sigma,\mu}\| ^2\leq \varepsilon \| \Delta u_{\sigma,\mu}\| ^2 +c\| u_{\sigma,\mu}\| ^2.$ Hence $$\int_{M}\frac{| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }} dv_{g}\leq C( 1+\varepsilon ) \| \Delta u_{\sigma,\mu}\| ^2+A( \varepsilon ) \| u_{\sigma,\mu}\| ^2 \label{14}$$ Now if $K(n,2,\sigma )$ denotes the best constant in inequality \eqref{14} we obtain that for any $\varepsilon >0$, $$\int_{M}\frac{| \nabla u_{\sigma,\mu}| ^2}{\rho ^{\sigma }} dv_{g}\leq ( K(n,2,\sigma )^2+\varepsilon ) \| \Delta u_{\sigma,\mu}\| ^2+A( \varepsilon ,\sigma ) \| u_{\sigma,\mu}\| ^2. \label{15}$$ By inequalities \eqref{11}, \eqref{13} and \eqref{15}, we have \begin{align*} \nu _{1,\sigma ,\mu } &\geq ( 1+a^{-}\max ( K(n,2,\sigma ),A(\varepsilon ,\sigma ) ) \\ &\quad +b^{-}\max ( K(n,2,\mu ),A( \varepsilon ,\mu ) ) ) ( \| \Delta u_{\sigma ,\mu }\| ^2+\| u_{\sigma ,\mu }\| ^2) \end{align*} So if $1+a^{-}\max ( K(n,2,\sigma ),A( \varepsilon ,\sigma ) ) +b^{-}\max ( K(n,2,\mu ),A( \varepsilon ,\mu )) >0$ then we obtain $\lim_{\sigma ,\mu }( u_{\sigma ,\mu }) =0$ and $\| u_{\sigma ,\mu }\| =1$ a contradiction. The reflexivity of $H_2^2(M)$ and the compactness of the embedding $H_2^2(M)\subset H_p^{k}(M)$ ($k=0,1$; $p1$ we put $I_p^q=\int_{0}^{+\infty }\frac{t^q}{(1+t)^{p}}dt\,.$ The following relations are immediate $I_{p+1}^q=\frac{p-q-1}{p}I_p^q, \quad I_{p+1}^{q+1}= \frac{q+1}{p-q-1}I_{p+1}^q.$ \subsection{Application to compact Riemannian manifolds of dimension $n>6$} \begin{theorem}\label{thm5} Let $( M,g)$ be a compact Riemannian manifold of dimension $n>6$. Suppose that at a point $x_0$ where $f$ attains its maximum the following condition $\frac{\Delta f(x_0)}{f( x_0) }<\frac{1}{3}\Big( \frac{ (n-1)n( n^2+4n-20) }{( n^2-4) ( n-4) ( n-6) }\frac{1}{( 1+\| a\| _r+\| b\| _s) ^{n/4}}-1\Big) S_{g}( x_0)$ holds. Then \eqref{1} has a non trivial solution with energy $J_{\lambda }(u)<\frac{1}{K_0^{n/4}( \max_{x\in M}f(x)) ^{\frac{n}{4}-1}}.$ \end{theorem} \begin{proof} The proof of Theorem \ref{thm5} reduces to show that the condition \eqref{C1} of Theorem \ref{thm3} is satisfied and since by Lemma \ref{lem0} there is a $t_0>0$ such that $t_0u_{\epsilon }\in M_{\lambda }$ for sufficiently small $\lambda$, so it suffices to show that $\sup_{t>0}J_{\lambda }( tu_{\epsilon }) <\frac{1}{K_0^{ \frac{n}{4}}( \max_{x\in M}f(x)) ^{\frac{n}{4}-1}}.$ To compute the term $\int_{M}f(x)| u_{\epsilon }(x)|^{N}dv_{g}$, we need the following Taylor's expansion of $f$ at the point $x_0$ $f(x)=f(x_0)+\frac{\partial ^2f(x_0)}{2\partial y^{i}\partial y^{j}}y^{i}y^{j}+o(\rho ^2)$ and also that of the Riemannian measure $dv_{g}=1-\frac{1}{6}R_{ij}(x_0)y^{i}y^{j}+o(\rho ^2)$ where $R_{ij}(x_0)$ denotes the Ricci tensor at $x_0$. The expression of $\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g}$ is well known (see for example \cite{11} ) and is given in case $n>6$ by $\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g}=\frac{\theta ^{-n}}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\Big( 1-( \frac{\Delta f(x_0)}{2(n-2)f(x_0)}+\frac{S_{g}(x_0)}{ 6(n-2)})\epsilon ^2+o(\epsilon ^2)\Big)$ where $K_0$ is given by \eqref{K} and $\omega _{n}=2^{n-1}I_{n}^{\frac{n}{2}-1}\omega _{n-1}$ and $\omega _{n}$ is the volume of $S^{n}$, the standard unit sphere of $R^{n+1}$ endowed with its round metric. Now the restriction of $| \frac{\partial u_{\epsilon }}{\partial \rho }|$ to the geodesic ball $B(x_0,\delta )$ is computed as follows $| \frac{\partial u_{\epsilon }}{\partial \rho }| _{B(x_0,\delta )}=| \nabla u_{\epsilon }| =\theta ^{-2}(n-4)( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}) ^{ \frac{n-4}{8}}\frac{\rho }{(( \frac{\rho }{\theta }) ^2+\epsilon ^2)^{\frac{n-2}{2}}}$ and Since $a\in L^{r}(M)$ with $r>n/2$ we have \begin{align*} \int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }| ^2dv_{g} &\leq \theta ^{-4}(n-4)^2\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4} }{f(x_0)}\Big) ^{\frac{n-4}{4}}\| a\| _r\omega _{n-1}^{1-\frac{1}{r}} \\ &\quad \times \Big( \int_{0}^{\delta }\frac{\rho ^{\frac{2r}{r-1}+n-1}}{(( \frac{\rho }{\theta }) ^2+\epsilon ^2)^{\frac{( n-2) r}{ r-1}}}\Big( \int_{S(\rho )}\sqrt{| g(x)| }d\Omega \Big) d\rho \Big) ^{\frac{r-1}{r}} \end{align*} Since $\int_{S(\rho )}\sqrt{| g(x)| }d\Omega =\omega _{n-1}\Big( 1-\frac{S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big)$ we obtain \begin{align*} \int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g} &\leq \theta ^{-4}(n-4)^2( \frac{(n-4)n(n^2-4)\epsilon ^{4} }{f(x_0)}) ^{\frac{n-4}{4}}\| a\| _r\omega _{n-1}^{1-\frac{1}{r}} \\ &\quad \times \Big( \int_{0}^{\delta }\frac{\rho ^{\frac{2r}{r-1}+n-1}}{(( \rho \theta ) ^2+\epsilon ^2)^{\frac{( n-2) r}{r-1}}} d\rho \Big( 1-\frac{S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big) \Big) ^{\frac{r-1}{r}} \end{align*} and by the following change of variable $t=(\frac{\rho \theta }{\epsilon })^2\quad\text{ i.e. } \rho =\frac{\epsilon }{\theta }\sqrt{t}$ we obtain \begin{align*} &\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g}\\ &\leq \theta ^{-n\frac{r}{r-1}}(n-4)^2\Big( \frac{ (n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4} }\| a\| _r\omega _{n-1}^{1-\frac{1}{r}}\epsilon ^{-( n-4) +2-\frac{n}{r}} \\ &\quad \times \Big( \int_{0}^{(\frac{\delta \theta }{\epsilon })^2}\frac{t^{ \frac{n-2}{2}+\frac{r}{r-1}}}{(t+1)^{\frac{( n-2) r}{r-1}}}dt- \frac{S_{g}(x_0)}{6n}\theta ^{-2}\epsilon ^2\int_{0}^{(\frac{\delta \theta }{\epsilon })^2}\frac{t^{\frac{n}{2}+\frac{r}{r-1}}}{(t+1)^{\frac{ ( n-2) r}{r-1}}}dt+o(\epsilon ^2)\Big) ^{\frac{r-1}{r}}. \end{align*} Letting $\epsilon \to 0$ we obtain \begin{align*} &\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g} \\ &\leq 2^{-1+\frac{1}{r}}\theta ^{-n( 1-\frac{1}{r}) }(n-4)^2( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}) ^{ \frac{n-4}{4}}\| a\| _r\omega _{n-1}^{1-\frac{1}{r} }\epsilon ^{-( n-4) +2-\frac{n}{r}} \\ &\quad \times ( I_{\frac{( n-2) r}{r-1}}^{\frac{n-2}{2}+\frac{r}{r-1 }}-\theta ^{-2}\frac{S_{g}(x_0)}{6n}I_{\frac{( n-2) r}{r-1} }^{\frac{n}{2}+\frac{r}{r-1}}\epsilon ^2+o(\epsilon ^2)) ^{\frac{ r-1}{r}}. \end{align*} Then \begin{align*} &\int_{B(x_0,\delta )}a(x)| \nabla u_{\epsilon }|^2dv_{g}\\ &\leq 2^{-1+\frac{1}{r}}\theta ^{-n\frac{r}{r-1}}(n-4)^2\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4} }\| a\| _r\omega _{n-1}^{1-\frac{1}{r}}\epsilon ^{\epsilon ^{-( n-4) +2-\frac{n}{r}}} \\ &\quad \times I_{\frac{( n-2) r}{r-1}}^{1+\frac{n-2}{2}.\frac{r-1}{r}} \Big[ 1-\frac{r-1}{r}\theta ^2\frac{S_{g}(x_0)}{6n}I_{\frac{( n-2) r}{r-1}}^{\frac{n}{2}+\frac{r}{r-1}}I_{\frac{( n-2) r}{ r-1}}^{-\frac{n-2}{2}-\frac{r}{r-1}}\epsilon ^2+o(\epsilon ^2)\Big]. \end{align*} It remains to compute the integral $\int_{B(x_0,2\delta )-B(x_0,\delta )}a(x)| \nabla u_{\epsilon }| ^2dv_{g}$. First we remark that $\big| \int_{(\frac{\delta \theta }{\epsilon })^2}^{(\frac{2\delta \theta }{\epsilon })^2}h(t)\frac{t^q}{(t+1)^{p}}dt\big| \leq C( \frac{1}{\epsilon }) ^{2(q-p+1)}=C\epsilon ^{2(p-q-1)}$ and since $p-q=n-4\geq 3$, we obtain $\int_{(\frac{\delta \theta }{\epsilon })^2}^{(\frac{2\delta \theta }{ \epsilon })^2}h(t)\frac{i^q}{(t+1)^{p}}dt=o(\epsilon ^2)$ and then $$\int_{B(x_0,2\delta )-B(x_0,\delta )}a(x)| \nabla u_{\epsilon }| ^2dv_{g}=o(\epsilon ^2). \label{19}$$ Finally we obtain \begin{align*} &\int_{M}a(x)| \nabla u_{\epsilon }| ^2dv_{g}\\ &\leq 2^{-1+ \frac{1}{r}}\theta ^{-n\frac{r}{r-1}}(n-4)^2\Big( \frac{ (n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4} }\| a\| _r\omega _{n-1}^{1-\frac{1}{r}}\epsilon ^{-( n-4) +2-\frac{n}{r}} \\ &\quad \times \Big( I_{\frac{( n-2) r}{r-1}}^{1+\frac{n-2}{2}.\frac{r-1 }{r}}+o(\epsilon ^2)\Big) . \end{align*} Letting $$A=K_0^{n/4}\frac{(n-4)^{\frac{n}{4}+1}\times ( \omega _{n-1}) ^{\frac{r-1}{r}}}{2^{\frac{r-1}{r}}}(n(n^2-4))^{\frac{n-4}{4} }\Big( I_{\frac{( n-2) r}{r-1}}^{\frac{n-2}{2}+\frac{r}{r-1} }\Big) ^{\frac{r-1}{r}} \label{20}$$ we obtain $\int_{M}a(x)| \nabla u_{\epsilon }| ^2dv_{g}\leq \epsilon ^{2-\frac{n}{r}}\theta ^{-n\frac{r}{r-1}} \frac{A}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\| a\|_r( 1+o(\epsilon ^2)) .$ Now we compute $\int_{M}b(x)u_{\epsilon }^2dv_{g}=\int_{B(x_0,\delta )}b(x)u_{\epsilon }^2dv_{g}+\int_{B(x_0,2\delta )-B(x_0,\delta )}b(x)u_{\epsilon }^2dv_{g}$ and since $b\in L^{s}(M)$ with $s>\frac{n}{4}$, we have $\int_{M}b(x)u_{\epsilon }^2dv_{g}\leq \| b\| _s\| u_{\epsilon }\| _{\frac{2s}{s-1}}^2.$ Independently, \begin{align*} \| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta )}^2 &=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{ \frac{n-4}{4}}\\ &\quad\times \Big( \int_{0}^{\delta }\frac{\rho ^{n-1}}{(( \rho \theta ) ^2+\epsilon ^2)^{\frac{( n-4) s}{(s-1)}}} \Big( \int_{S(r)}\sqrt{| g(x)| }d\Omega \Big) dr\Big) ^{\frac{s-1}{s}} \end{align*} and $\int_{S(r)}\sqrt{| g(x)| }d\Omega =\omega _{n-1}\Big( 1-\frac{S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big).$ Consequently, \begin{align*} \| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta)}^2 &=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}}\\ \omega _{n-1}^{\frac{s-1}{s}} &\quad\times \Big(\int_{0}^{\delta }\frac{\rho ^{n-1}}{(( \rho \theta ) ^2+\epsilon ^2)^{\frac{( n-4) s}{(s-1)}}}\Big( 1-\frac{ S_{g}(x_0)}{6n}\rho ^2+o(\rho ^2)\Big) d\rho \Big) ^{\frac{s-1}{s}}. \end{align*} And putting $t=(\rho \theta/\epsilon )^2$ , we obtain \begin{align*} \| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta)}^2 &=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}} ( \omega _{n-1}) ^{\frac{s-1}{s}}\epsilon ^{-n+4+4- \frac{n}{s}}\\ &\quad\times \Big(\frac{\epsilon ^{n}\theta ^{-n}}{2}\int_{0}^{(\frac{\delta \theta }{ \epsilon })^2}\frac{t^{\frac{n}{2}-1}}{(t+1)^{\frac{( n-4) s}{ (s-1)}}}dt\\ &\quad -\frac{\theta ^{-n-2}S_{g}(x_0)}{12n}\epsilon ^{n+2}\int_{0}^{(\frac{\delta \theta }{\epsilon })^2}\frac{t^{\frac{n}{2}} }{(t+1)^{\frac{( n-4) s}{(s-1)}}}dt+o(\epsilon ^{n+2})\Big) ^{\frac{s-1}{s}}. \end{align*} Letting $\epsilon \to 0$, we obtain \begin{align*} \| u_{\epsilon }\| _{\frac{2s}{s-1},B( x_0,\delta ) }^2 &=\Big(\frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)} \Big) ^{\frac{n-4}{4}}( \omega _{n-1}) ^{\frac{s-1}{s}}\epsilon ^{-n+4+4-\frac{n}{s}}\\ &\quad \times \theta ^{-n\frac{s}{s-1}}(\frac{\epsilon ^{n}}{2})^{\frac{s-1}{s} }\Big( \int_{0}^{+\infty }\frac{t^{\frac{n}{2}}}{(t+1)^{\frac{( n-4) s}{(s-1)}}}dt\\ &\quad -\frac{S_{g}(x_0)}{12n}\epsilon ^2\theta ^{-2}\int_{0}^{+\infty }\frac{t^{\frac{n}{2}+1}}{(t+1)^{\frac{( n-4) s}{(s-1)}}}dt+o(\epsilon ^2)\Big) ^{\frac{s-1}{s}}. \end{align*} Hence \begin{align*} &\| u_{\epsilon }\| _{\frac{2s}{s-1},B(x_0,\delta)}^2\\ &=\Big( \frac{(n-4)n(n^2-4)\epsilon ^{4}}{f(x_0)}\Big) ^{\frac{n-4}{4}} ( \omega _{n-1}) ^{\frac{s-1}{s}}\epsilon ^{-n+4+4- \frac{n}{s}}\theta ^{-n\frac{s}{s-1}}(\frac{\epsilon ^{n}}{2})^{\frac{s-1}{s} }\\ &\quad\times \Big( \int_{0}^{+\infty }\frac{t^{\frac{n}{2}}}{(t+1)^{\frac{( n-4) s}{(s-1)}}}dt-\theta ^{-2}\frac{S_{g}(x_0)}{12n}\epsilon ^2\int_{0}^{+\infty }\frac{t^{\frac{n}{2}+1}}{(t+1)^{\frac{( n-4) s}{(s-1)}}}dt+o(\epsilon ^2)\Big) ^{\frac{s-1}{s}}, \end{align*} or \begin{align*} \| u_{\epsilon }\| _{\frac{2s}{s-1}}^2 &=\Big( \frac{ (n-4)n(n^2-4)}{f(x_0)}\Big) ^{\frac{n-4}{4}} \Big( \frac{\omega _{n-1}}{2}\Big) ^{\frac{s-1}{s}} \epsilon ^{4-\frac{n}{s}}\theta ^{-n\frac{s}{s-1}} \\ &\quad \times \Big[ ( I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2} }) ^{\frac{s-1}{s}}-\frac{\theta ^{-2}(s-1)S_{g}(x_0)}{12n\text{ }s}( I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2}}) ^{- \frac{1}{s}}I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2}+1}\epsilon ^2+o(\epsilon ^2)\Big] \end{align*} Finally, by the same method as in equality \eqref{19}, we obtain \begin{align*} &\int_{M}b(x)u_{\epsilon }^2dv_{g}\\ &\leq \| b\| _s( \frac{(n-4)n(n^2-4)}{f(x_0)}) ^{\frac{n-4}{4}}(\frac{\omega _{n-1}}{2})^{\frac{s-1}{s}}\epsilon ^{4-\frac{n}{s}}\theta ^{-n\frac{s}{s-1}} \Big( \Big( I_{\frac{( n-4) s}{(s-1)}}^{\frac{n}{2} }\Big) ^{\frac{s-1}{s}}+o(\epsilon ^2)\Big). \end{align*} Putting $$B=K_0^{n/4}((n-4)n(n^2-4))^{\frac{n-4}{4}}(\frac{\omega _{n-1}}{2})^{\frac{s-1}{s}}\Big( I_{\frac{( n-4) s}{(s-1)}}^{ \frac{n}{2}}\Big) ^{\frac{s-1}{s}} \label{21}$$ we obtain $\int_{M}b(x)u_{\epsilon }^2dv_{g}\leq \epsilon ^{4-\frac{n}{s}}\theta ^{-n \frac{s}{s-1}}\frac{\| b\| _sB}{\text{ }K_0^{\frac{ n}{4}}(f(x_0))^{\frac{n-4}{4}}}( 1+o(\epsilon ^2)) .$ The computation of $\int_{M}( \Delta u_{\epsilon }) ^2dv_{g}$ is well known see for example (\cite{11})\ and is given by $\int_{M}( \Delta u_{\epsilon }) ^2dv_{g}=\frac{\theta ^{-n}}{ K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\Big( 1-\frac{ n^2+4n-20}{6(n^2-4)(n-6)}S_{g}(x_0)\epsilon ^2+o(\epsilon ^2)\Big) .$ Summarizing, we obtain \begin{align*} &\int_{M}( \Delta u_{\epsilon }) ^2-a(x)| \nabla u_{\epsilon }| ^2+b(x)u_{\epsilon }^2dv_{g}\\ &\leq \frac{\theta ^{-n}}{K_0^{n/4}f(x_0)^{\frac{n-4}{4}}} \Big( 1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n}{r-1}}A\| a\| _r+\epsilon ^{4-\frac{n}{s}}\theta ^{-\frac{n}{s-1} }B\| b\| _s\\ &\quad -\frac{n^2+4n-20}{6(n^2-4)(n-6)} S_{g}(x_0)\epsilon ^2 +o(\epsilon ^2)\Big) . \end{align*} Now, we have \begin{align*} J_{\lambda }( tu_{\epsilon }) &\leq J_0( tu_{\epsilon}) =\frac{t^2}{2}\| u_{\epsilon }\| ^2-\frac{t^{N} }{N}\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g} \\ &\leq \frac{\theta ^{-n}}{K_0^{n/4}f(x_0)^{\frac{n-4}{4} }}\Big\{ \frac{1}{2}t^2( 1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n }{r-1}}A\| a\| _r+\epsilon ^{4-\frac{n}{s}}\theta ^{- \frac{n}{s-1}}B\| b\| _s) -\frac{t^{N}}{N}\\ &\quad +\Big[ \Big( \frac{\Delta f(x_0)}{2( n-2) f(x_0)}+ \frac{S_{g}( x_0) }{6( n-1) }\Big) \frac{t^{N}}{N}- \frac{1}{2}t^2\frac{n^2+4n-20}{6( n^2-4) ( n-6) } S_{g}( x_0) \Big] \epsilon ^2\Big\}\\ &\quad +o( \epsilon ^2) \end{align*} and letting $\epsilon$ be small enough so that $1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n}{r-1}}A\| a\| _r+\epsilon ^{4-\frac{n}{s}}\theta ^{-\frac{n}{s-1}}B\| b\| _s\leq ( 1+\| a\| _r+\| b\| _s) ^{\frac{4}{n}}$ and since the function $\varphi (t)=\alpha \frac{t^2}{2}-\frac{t^{N}}{N}$, with $\alpha >0$ and $t>0$, attains its maximum at $t_0=\alpha ^{\frac{1}{N-2}}$ and $\varphi (t_0)=\frac{2}{n}\alpha ^{n/4}.$ Consequently, \begin{align*} J_{\lambda }( tu_{\epsilon }) &\leq \frac{2\theta ^{-n}}{nK_0^{n/4}f(x_0)^{\frac{n-4}{4}}} \Big\{ 1+\|a\| _r+\| b\| _s +\Big[ \Big( \frac{\Delta f(x_0)}{2( n-2) f(x_0)}+ \frac{S_{g}( x_0) }{6( n-1) }\Big) \frac{t_0^{N}}{N}\\ &\quad -\frac{1}{2}t_0^2\frac{n^2+4n-20}{6( n^2-4) ( n-6) }S_{g}( x_0) \Big] \epsilon ^2\Big\} +o( \epsilon ^2) . \end{align*} Taking into account the value of $\theta$ and putting $R(t)=\Big( \frac{\Delta f(x_0)}{2( n-2) f(x_0)}+\frac{ S_{g}( x_0) }{6( n-1) }\Big) \frac{t^{N}}{N}-\frac{ 1}{2}\frac{n^2+4n-20}{6( n^2-4) ( n-6) } S_{g}( x_0) t^2$ we obtain $\sup_{t\geq 0}J_{\lambda }( tu_{\epsilon }) <\frac{2}{nK_0^{n/4}( \max_{x\in M}f(x)) ^{\frac{n}{4}-1}}$ provided that $R(t_0)<0$; i.e., $\frac{\Delta f(x_0)}{f( x_0) }< \Big( \frac{n(n^2+4n-20) }{3( n+2) ( n-4) ( n-6) } \frac{1}{( 1+\| a\| _r+\| b\|_s) ^{n/4}}-\frac{n-2}{3( n-1)}\Big) S_{g}( x_0) .$ Which completes the proof. \end{proof} \subsubsection{Application to compact Riemannian manifolds of dimension $n=6$} \begin{theorem} In case $n=6$, we suppose that at a point $x_0$ where $f$ attains its maximum $S_{g}( x_0) >0$. Then the equation \eqref{1} has a non trivial solution. \end{theorem} \begin{proof} The same calculations as in case $n>6$ gives us $\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g} =\frac{\theta ^{-n}}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}} \Big( 1-\Big( \frac{\Delta f(x_0)}{2(n-2)f(x_0)}+\frac{S_{g}(x_0)}{ 6(n-2)}\Big)\epsilon ^2+o(\epsilon ^2)\Big) .$ Also, we have $\int_{M}a(x)| \nabla u_{\epsilon }| ^2dv_{g}\leq \frac{ \| a\| _rA}{\text{ }K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\epsilon ^{2-\frac{n}{r}\theta ^{-\frac{r}{r-1}}}( 1+o( \epsilon ^2) )$ and $\int_{M}b(x)u_{\epsilon }^2dv_{g}\leq \frac{\| b\| _sB}{ K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}\epsilon ^{4-\frac{n }{s}}\theta ^{-\frac{s}{s-1}}+( 1+o( \epsilon ^2) ).$ where $A$ and $B$ are given by \eqref{20} and \eqref{21} respectively for $n=6$. The computations of the term $\int_{M}( \Delta u_{\epsilon}) ^2dv_{g}$ are well known (see for example \cite{11}) \begin{align*} &\int_{M}( \Delta u_{\epsilon }) ^2dv(g)\\ &=\theta ^{-n}(n-4)^2\Big( \frac{(n-4)n(n^2-4)}{f(x_0)}\Big)^{\frac{n-4}{4}}\frac{\omega _{n-1}}{2}\\ &\quad \times \Big( \frac{n(n+2)(n-2)}{(n-4)}I_{n}^{\frac{n}{2}-1}-\frac{2}{n} \theta ^{-2}S_{g}(x_0)\epsilon ^2\log (\frac{1}{\epsilon ^2} )+O(\epsilon ^2)\Big) . \end{align*} $\int_{M}( \Delta u_{\epsilon }) ^2dv_{g} =\frac{\theta ^{-n}}{K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}} \Big( 1-\frac{ 2( n-4) }{n^2(n^2-4)I_{n}^{\frac{n}{2}-1}}S_{g}(x_0)\epsilon ^2 \log ( \frac{1}{\epsilon ^2}) +O(\epsilon^2)\Big) .$ Now summarizing and letting $\epsilon$ so that $1+\epsilon ^{2-\frac{n}{r}}\theta ^{-\frac{n}{r-1}}A\| b\| _s+\epsilon ^{4-\frac{n}{s}}\theta ^{-\frac{n}{s-1}}B\| a\| _r\leq ( 1+\| a\| _r+\| b\| _s) ^{\frac{4}{n}}$ we obtain \begin{align*} J_{\lambda }( u_{\epsilon }) &\leq \frac{1}{2}\|u_{\epsilon }\| ^2-\frac{1}{N}\int_{M}f(x)| u_{\epsilon }(x)| ^{N}dv_{g} \\ &\leq \frac{\theta ^{-n}}{\text{ }K_0^{n/4}(f(x_0))^{ \frac{n-4}{4}}}\Big[ \frac{t^2}{2}( 1+\| a\| _r+\| b\| _s) ^{1-\frac{4}{n}}-\frac{t^{N}}{N} \\ &\quad -\frac{n-4}{n^2( n^2-4) I_{n}^{\frac{n}{2}-1}}\theta ^{-2}S_{g}(x_0)t^2\epsilon ^2\log ( \frac{1}{\epsilon ^2} ) \Big] +O(\epsilon ^2). \end{align*} The same arguments as in the case $n>6$ allow us to infer that $\max_{t\geq 0}J_{\lambda }( tu_{\epsilon }) <\frac{2}{n\text{ } K_0^{n/4}(f(x_0))^{\frac{n-4}{4}}}$ if $S_{g}(x_0)>0$. 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