Variational theory for the resonant T-curvature equation

Abstract

In this paper, we study the resonant prescribed  T-curvature problem on a compact  4-dimensional Riemannian manifold with boundary. We derive sharp energy and gradient estimates of the associated Euler-Lagrange functional to characterize the critical points at infinity of the associated variational problem under a non-degeneracy on a naturally associated Hamiltonian function. Using this, we derive a Morse type lemma at infinity around the critical points at infinity. Using the Morse lemma at infinity, we prove new existence results of Morse theoretical type. Combining the Morse lemma at infinity and the Liouville version of the Barycenter technique of Bahri–Coron (Commun Pure Appl Math 41–3:253–294, 1988) developed in Ndiaye (Adv Math 277(277):56–99, 2015), we prove new existence results under a topological hypothesis on the boundary of the underlying manifold, the selection map at infinity, and the entry and exit sets at infinity.

Appendix

Lemma 5.1

Assuming that  $\epsilon$  is positive and small,  $a\in \partial M$  and  $\lambda \ge \frac{1}{\epsilon }$, then

• (1)
  $$\begin{aligned} \varphi _{a, \lambda }(\cdot )={\hat{\delta }}_{a, \lambda }(\cdot )+\log \frac{\lambda }{2}+H(a, \cdot )+\frac{1}{2\lambda ^2}\Delta _{{\hat{g}}_{a}}H(a, \cdot )+O\left( \frac{1}{\lambda ^3}\right) \,\,\,\textrm{on}\,\,\,\partial M \end{aligned}$$
• (2)
  $$\begin{aligned} \lambda \frac{\partial \varphi _{a, \lambda }(\cdot )}{\partial \lambda }=\frac{2}{1+\lambda ^2\chi _{\varrho }^2(d_{{\hat{g}}_{a}}(a, \cdot ))}-\frac{1}{\lambda ^2}\Delta _{{\hat{g}}_{a}}H(a, \cdot )+O\left( \frac{1}{\lambda ^3}\right) \,\,\,\textrm{on}\,\,\,\partial M, \end{aligned}$$
• (3)
  $$\begin{aligned} \frac{1}{\lambda }\frac{\partial \varphi _{a, \lambda }(\cdot )}{\partial a}= & {} \frac{\chi _{\varrho }(d_{{\hat{g}}_{a}}(a, \cdot ))\chi _{\varrho }^{'}((d_{{\hat{g}}_{a}}(a, \cdot ))}{d_{{\hat{g}}_{a}}(a, \cdot )}\frac{2\lambda exp_a^{-1}(\cdot )}{1+\lambda ^2\chi _{\varrho }^2(d_{{\hat{g}}_a}(a, \cdot ))}+\frac{1}{\lambda }\frac{\partial H(a, \cdot )}{\partial a}\\{} & {} +O\left( \frac{1}{\lambda ^3}\right) ; \textrm{on}\,\,\,\partial M, \end{aligned}$$

  where  O(1) means $O_{a, \lambda , \epsilon }(1)$ and for it meaning see Sect. 2.

Lemma 5.2

Assuming that  $\epsilon$  is small and d positive,  $a\in M$, $\lambda \ge \frac{1}{\epsilon }$, and  $0<2\eta <\varrho$  with  $\varrho$  as in (43), then there holds

$$\begin{aligned}{} & {} \varphi _{a, \lambda }(\cdot )=G(a, \cdot )+\frac{1}{2\lambda ^2}\Delta _{{\hat{g}}_{a}}G(a, \cdot )+O\left( \frac{1}{\lambda ^3}\right) \,\,\textrm{on} \,\,\,\partial M\setminus B^{a}_{a}(\eta ), \\{} & {} \lambda \frac{\partial \varphi _{a, \lambda }(\cdot )}{\partial \lambda }=-\frac{1}{\lambda ^2}\Delta _{{\hat{g}}_{a}}G(a, \cdot )+O\left( \frac{1}{\lambda ^3}\right) \,\,\textrm{on} \,\,\,\partial M\setminus B^{a}_{a}(\eta ), \end{aligned}$$

and

$$\begin{aligned} \frac{1}{\lambda }\frac{\partial \varphi _{a, \lambda }(\cdot )}{\partial a}=\frac{1}{\lambda }\frac{\partial G(a, \cdot )}{\partial a}+O\left( \frac{1}{\lambda ^3}\right) \,\,\textrm{on} \,\,\,\partial M\setminus B^{a}_{a}(\eta ), \end{aligned}$$

where  O(1)  means  $O_{a, \lambda , \epsilon }(1)$  and for it meaning see Sect. 2.

Lemma 5.3

Assuming that  $\epsilon$  is small and positive, $a\in \partial M$ and $\lambda \ge \frac{1}{\epsilon }$, then there holds

$$\begin{aligned}{} & {} \mathbb {P}_g^{4, 3}\left( \varphi _{a, \lambda }, \,\varphi _{a, \lambda }\right) =16\pi ^2\log \lambda -8\pi ^2C_0+8\pi ^2 H(a, a)+\frac{8\pi ^2}{\lambda ^2}\Delta _{{\hat{g}}_a} H(a, a)\\{} & {} \quad +O\left( \frac{1}{\lambda ^3}\right) , \\{} & {} \mathbb {P}_g^{4, 3}\left( \varphi _{a, \lambda },\, \lambda \frac{\varphi _{a, \lambda }}{\partial \lambda } \right) =8\pi ^2-\frac{8\pi ^2}{\lambda ^2}\Delta _{{\hat{g}}_a}H(a, a)+O\left( \frac{1}{\lambda ^3}\right) , \\{} & {} \mathbb {P}_g^{4, 3}\left( \varphi _{a, \lambda }, \,\frac{1}{\lambda }\frac{\varphi _{a, \lambda }}{\partial a}\right) =\frac{8\pi ^2}{\lambda }\frac{\partial H(a, a)}{\partial a}+O\left( \frac{1}{\lambda ^3}\right) , \end{aligned}$$

where  $C_0$  is a positive constant, O(1)  means  $O_{a, \lambda , \epsilon }(1)$  and for its meaning see Sect. 2.

Lemma 5.4

Assuming that $\epsilon$ is small and positive $a_i, a_j\in \partial M$,  $d_{{\hat{g}}}(a_i, a_j)\ge 4\overline{C}\eta$, $0<2\eta <\varrho$, $\frac{1}{\Lambda }\le \frac{\lambda _i}{\lambda _j}\le \Lambda$, and $\lambda _i, \lambda _j\ge \frac{1}{\epsilon }$, $\overline{C}$ as in (41), and $\varrho$ as in (43), then there hold

$$\begin{aligned}{} & {} \begin{aligned} \mathbb {P}_g^{4, 3}\left( \varphi _{a_i, \lambda _i}, \,\varphi _{a_j, \lambda _j}\right) =&8\pi ^2 G (a_j, a_i)+\frac{4\pi ^2}{\lambda _i^2}\Delta _{{\hat{g}}_{a_i}} G(a_i, a_j)+\frac{4\pi ^2}{\lambda _j^2}\Delta _{{\hat{g}}_{a_j}} G(a_j, a_i)\\&+O\left( \frac{1}{\lambda ^3_i}+\frac{1}{\lambda _j^3}\right) , \end{aligned} \\{} & {} \mathbb {P}_g^{4, 3}\left( \varphi _{a_i, \lambda _i}, \,\lambda _j\frac{\partial \varphi _{a_j, \lambda _j}}{\partial \lambda _j}\right) =-\frac{8\pi ^2}{\lambda _j^2}\Delta _{{\hat{g}}_{a_j}}G(a_j, a_i)+O\left( \frac{1}{\lambda ^3_j}\right) , \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}_g^{4, 3}\left( \varphi _{a_i, \lambda _i}, \,\frac{1}{\lambda _j}\frac{\partial \varphi _{a_j, \lambda _j}}{\partial a_j}\right) = \frac{8\pi ^2}{\lambda _j}\frac{\partial G(a_j, a_i)}{\partial a_j}+O\left( \frac{1}{\lambda ^3_j}\right) , \end{aligned}$$

where  O(1)  means here  $O_{A, {\bar{\lambda }}, \epsilon }(1)$  with  $A=(a_i, a_j)$  and  ${\bar{\lambda }}=(\lambda _i, \lambda _j)$  and for the meaning of $O_{A, {\bar{\lambda }}, \epsilon }(1)$, see Sect. 2.

Lemma 5.5

Assuming that $\epsilon >0$ is very small, we have that for $a\in \partial M$, $\lambda \ge \frac{1}{\epsilon }$, there holds

$$\begin{aligned}{} & {} \left\| \lambda \frac{\partial \varphi _{a, \lambda }}{\partial \lambda }\right\| _{\mathbb {P}^{4, 3}}={\tilde{O}}(1), \end{aligned}$$

(119)

$$\begin{aligned}{} & {} \left\| \frac{1}{\lambda }\frac{\partial \varphi _{a, \lambda }}{\partial a}\right\| _{\mathbb {P}^{4, 3}}={\tilde{O}}(1), \end{aligned}$$

(120)

and

$$\begin{aligned} \left\| \frac{1}{\sqrt{\log \lambda }}\varphi _{a, \lambda }\right\| _{\mathbb {P}^{4, 3}}={\tilde{O}}(1), \end{aligned}$$

(121)

where here ${\tilde{O}}(1)$ means bounded by positive constants form below and above independent of $\epsilon$, a, and $\lambda$.

Lemma 5.6

• (1) If $\epsilon$ is small and positive, $a \in \partial M$, $p\in \mathbb {N}^*$, and $\lambda \ge \frac{1}{\epsilon }$, then there holds

  $$\begin{aligned} C^{-1}\lambda ^{6p-3}\le \oint _{\partial M}e^{3p\varphi _{a, \lambda }}dS_g\le C\lambda ^{6p-3}, \end{aligned}$$

  (122)

  where C is independent of a, $\lambda$, and $\epsilon$.

• (2) If $\epsilon$ is positive and small, $a_i, a_j \in \partial M$, $\lambda \ge \frac{1}{\epsilon }$ and $\lambda d_{{\hat{g}}}(a_i,a_j) \ge 4\overline{C} R$, then we have

  $$\begin{aligned} \mathbb {P}_g^{4, 3} (\varphi _{a_i,\lambda }, \,\varphi _{a_j,\lambda })\, \le \,8\pi ^2 G(a_i,a_j) \, + \, O(1), \end{aligned}$$

  (123)

  where  O(1)  means here  $O_{A, \lambda , \epsilon }(1)$  with  $A=(a_i, a_j)$, and for the meaning of $O_{A, \lambda , \epsilon }(1)$, see Sect. 2.

• (3) If $\epsilon$ is positive and small, $a_i, a_j \in \partial M$, $\lambda _i, \lambda _j\ge \frac{1}{\epsilon }$, $\frac{1}{\Lambda }\le \frac{\Lambda _i}{\lambda _j}\le \Lambda$ and $\lambda _id_{{\hat{g}}}(a_i,a_j) \ge 4\overline{C} R$, then we have

  $$\begin{aligned} \mathbb {P}_g^{4, 3} (\varphi _{a_i,\lambda }, \,\varphi _{a_j,\lambda }) \, \le \, 8\pi ^2 G(a_i,a_j) \, + \, O(1), \end{aligned}$$

  (124)

  where  O(1)  means here  $O_{A, {\bar{\lambda }}, \epsilon }(1)$  with  $A=(a_i, a_j)$  and  ${\bar{\lambda }}=(\lambda _i, \lambda _j)$  and for the meaning of $O_{A, {\bar{\lambda }}, \epsilon }(1)$, see Sect. 2.

Lemma 5.7

Let $p\in \mathbb {N}^*$, ${\hat{R}}$ be a large positive constant, $\epsilon$ be a small positive number, $\alpha _i\ge 0$, $i=1, \ldots , p$, $\sum _{i=1}^p\alpha _i=k$, $\lambda \ge \frac{1}{\epsilon }$ and $u = \sum _{i=1}^p \alpha _i \varphi _{a_i,\lambda }$. Assuming that there exist two positive integer $i, j\in \{1, \ldots , p\}$ with $i\ne j$ such that $\lambda d_{{\hat{g}}}(a_i,a_j) \le \frac{{\hat{R}}}{4\overline{C}}$, where $\overline{C}$ is as in (41), then we have

$$\begin{aligned} \mathcal {E}_g(u) \, \le \mathcal {E}_g(v) \, + \, O(\log {\hat{R}}), \end{aligned}$$

(125)

with

$$\begin{aligned} v:= \sum _{k \le p, k \ne i,j} \alpha _k \varphi _{a_k,\lambda } \, + (\alpha _i + \alpha _j) \varphi _{a_i,\lambda }. \end{aligned}$$

where here  O(1) stand for  $O_{{{\bar{\alpha }}}, A, \lambda , \epsilon }(1)$, with  ${{\bar{\alpha }}}=(\alpha _1, \ldots , \alpha _p)$  and $A=(a_1, \ldots , a_p)$, and for the meaning of  $O_{{{\bar{\alpha }}}, A, \lambda , \epsilon }(1)$, we refer the reader to Sect. 2.

Lemma 5.8

• (1) If $\epsilon$ is positive and small, $a_i, a_j\in \partial M$, $\lambda \ge \frac{1}{\epsilon }$ and $\lambda d_{{\hat{g}}}(a_i, a_j)\ge 4\overline{C} R$, then

  $$\begin{aligned} \varphi _{a_j, \lambda }(\cdot )=G(a_j,\cdot )+O(1)\,\,\,\text {in}\,\,B^{a_i}_{a_i}\left( \frac{R}{\lambda }\right) , \end{aligned}$$

  where here  O(1) means here  $O_{ A, \lambda , \epsilon }(1)$, with  $A=(a_i, a_j)$, and for the meaning of  $O_{ A, \lambda , \epsilon }(1)$, see Sect. 2.

• (2) If $\epsilon$ is positive and small, $a_i, a_j\in \partial M$, $\lambda _i, \lambda _j\ge \frac{1}{\epsilon }$, $\frac{1}{\Lambda }\le \frac{\Lambda _i}{\lambda _j}\le \Lambda$, and $\lambda _i d_{{\hat{g}}}(a_i, a_j)\ge 4\overline{C}R$, then

  $$\begin{aligned} \varphi _{a_j, \lambda _j}(\cdot )=G(a_j,\cdot )+O(1)\,\,\,\text {in}\,\,B^{a_i}_{a_i}\left( \frac{R}{\lambda _i}\right) , \end{aligned}$$

  where here  O(1) means here  $O_{ A, {\bar{\lambda }}, \epsilon }(1)$, with  $A=(a_i, a_j)$, ${\bar{\lambda }}=(\lambda _i, \lambda _j)$ and for the meaning of  $O_{ A, \lambda , \epsilon }(1)$, see Sect. 2.

Lemma 5.9

There exists  $\Gamma _0$  and  ${\tilde{\Lambda }}_0$ two large positive constant such that for every $a\in \partial M$, $\lambda \ge {\tilde{\Lambda }}_0$, and  $w\in F_{a, \lambda }:=\{w\in \mathcal {H}_{\frac{\partial }{\partial n}}, \overline{w}_{(Q, T)}=\left\langle \varphi _{a, \lambda }, w\right\rangle _{\mathbb {P}^{4, 3}}=\left\langle v_r, w\right\rangle _{\mathbb {P}^{4, 3}}=0,\,r=1,\ldots ,{\bar{k}}\}$, we have

$$\begin{aligned} \oint _{\partial M}e^{3{\hat{\delta }}_{a,\lambda }}w^2dV_{g_{a}}\le \Gamma _0\Vert w\Vert ^2_{\mathbb {P}^{4, 3}}. \end{aligned}$$

(126)

Lemma 5.10

Assuming that $\eta$  is a small positive real number with $0<2\eta <\varrho$ where $\varrho$ is as in (43), then there exists a small positive constant $c_0:=c_0(\eta )$ and $\Lambda _0:=\Lambda _0(\eta )$ such that for every $a_i\in \partial M$ concentrations points with $d_{{\hat{g}}}(a_i, a_j)\ge 4\overline{C}\eta$ where ${\bar{C}}$ is as in (41), for every  $\lambda _i>0$  concentrations parameters satisfying $\lambda _i\ge \Lambda _0$, with $i=1, \ldots , k$, and for every  $w\in E_{A, {\bar{\lambda }}}^*=\cap _{i=1}^k E^*_{a_i, \lambda _i}$ with $A:=(a_1, \ldots , a_k$), ${\bar{\lambda }}:= (\lambda _1, \ldots , \lambda _k)$ and $E^*_{a_i, \lambda _i}=\{w\in \mathcal {H}_{\frac{\partial }{\partial n}}: \,\,\left\langle \varphi _{a_i, \lambda _i}, w\right\rangle _{\mathbb {P}^{4, 3}}=\left\langle \frac{\partial \varphi _{a_i, \lambda _i}}{\partial \lambda _i}, w\right\rangle _{\mathbb {P}^{4, 3}}=\left\langle \frac{\partial \varphi _{a_i, \lambda _i}}{\partial a_i}, w\right\rangle _{\mathbb {P}^{4, 3}}=\overline{w}_{(Q, T)}=\left\langle v_r, w\right\rangle _{\mathbb {P}^{4, 3}}=0,\,r=1,\ldots ,{\bar{k}}\}$, there holds

$$\begin{aligned} \Vert w\Vert ^2_{\mathbb {P}^{4, 3}}- 6\sum _{i=1}^k\oint _{\partial M}e^{3{\hat{\delta }}_{a_i,\lambda _i}}w^2dS_{g_{a_i}}\ge c_0\Vert w\Vert ^2_{\mathbb {P}^{4, 3}}. \end{aligned}$$

(127)