Abstract
We study the blow-up behavior of minimizing sequences for the singular Moser–Trudinger functional on compact surfaces. Assuming non-existence of minimum points, we give an estimate for the infimum value of the functional. This result can be applied to give sharp Onofri-type inequalities on the sphere in the presence of at most two singularities.
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1 Introduction
Let \((\Sigma ,g)\) be a smooth, compact Riemannian surface; the standard Moser–Trudinger inequality (see [16, 22]) states that
where \(C(\Sigma ,g)\) is a constant depending only on \(\Sigma \) and \(g\), and the coefficient \(\frac{1}{16\pi }\) is optimal. A sharp version of (1) was proved by Onofri in [23] for the sphere endowed with the standard Euclidean metric \(g_0\). He identified the sharp value of \(C\) and the family of functions attaining equality, proving
with equality holding if and only if the metric \(e^{u}g\) has constant positive Gaussian curvature, or, equivalently, \(u=\log |\det d \varphi |+c\) with \(c\in {\mathbb {R}}\) and \(\varphi \) a conformal diffeomorphism of \(S^2\). Onofri’s inequality played an important role (see [12, 13]) in the variational approach to the equation
which is connected to the classical problem of prescribing the Gaussian curvature of \(S^2\). In this paper we will consider extensions of Onofri’s result in connection with the study of the more general equation
where \(K\in C^\infty (\Sigma )\) is a positive function, \(\rho >0\), \(p_1,\ldots , p_m\in \Sigma \) and \(\alpha _1,\ldots ,\alpha _m \in (-1,+\infty )\). This is known as the singular Liouville equation and arises in several problems in Riemannian geometry and mathematical physics. When \((\Sigma ,g)=(S^2,g_0)\) and \(\rho = 8\pi +4\pi \sum _{i=1}^m \alpha _i\), solutions of (3) provide metrics on \(S^2\) with prescribed Gaussian curvature \(K\) and conical singularities of angle \(2\pi (1+\alpha _i)\) (or of order \(\alpha _i\)) in \(p_i\), \(i=1,\ldots ,m\) (see for example [3, 14, 27]). Equation (3) also appears in the description of Abelian Chern–Simons vortices in superconductivity and Electroweak theory [17, 25]. We refer to [4, 9–11, 21], for some recent existence results. Liouville equations also have applications in the description of holomorphic curves in \({\mathbb {C}}{\mathbb {P}}^n\) [6, 8] and in the nonabelian Chern–Simons theory which might have applications in high temperature superconductivity (see [26] and references therein). Denoting by \(G_p\) the Green’s function at \(p\), namely the solution of
the change of variables
transforms (3) into
where
satisfies
with \(c_i>0\).
In [27], studying curvature functions for surfaces with conical singularities, Troyanov proved that if \(h\in C^\infty (\Sigma \backslash \{p_1,\ldots ,p_m\})\) is a positive function satisfying (6), then
The optimal constant \(C(\Sigma ,g,h)\) can be obtained by minimizing the functional
where \(\overline{\rho } =\displaystyle {\min \left\{ 1,1+\min _{1\le i\le m}\alpha _i\right\} } \). In this paper we will assume non-existence of minimum points for \(J_{\overline{\rho }}\) and exploit known blow-up results [1, 2, 5] to describe the behavior of a suitable minimizing sequence and compute \(\displaystyle {\inf _{H^1(\Sigma )}J_{\overline{\rho }}}\). The same technique was used by Ding, Jost, Li and Wang [15] to give an existence result for (3) in the regular case. From their proof it follows that if \(\alpha _i=0\) \(\forall \; i\) and if there is no minimum for \(J_{\overline{\rho }}\), then
where \(A(p)\) is the value in \(p\) of the regular part of \(G_p\). Here we extend this result to the general case proving:
Theorem 1.1
Assume that \(h\) satisfies (5) with \(K\in C^\infty (\Sigma )\), \(K>0\), \(\alpha _i \in (-1,+\infty )\backslash \{0\}\), and that there is no minimum point of \(J_{\overline{\rho }}\). If \(\alpha :=\displaystyle {\min _{1\le i\le m} \alpha _i<0}\), then
while if \(\displaystyle {\alpha >0}\)
In the last part of the paper we consider the case of the standard sphere with \(K\equiv 1\) and at most two singularities. When \(m=1\) a simple Kazdan–Warner type identity proves non-existence of solutions for (4). Thus, one can apply Theorem 1.1 to obtain the following sharp version of (7):
Theorem 1.2
If \(h=e^{-4\pi \alpha _1 G_{p_1}}\) with \(\alpha _1\ne 0\), then \(\forall \; u\in H^1(S^2)\)
The same non-existence argument works for \(m=2\), \(\min \{\alpha _1,\alpha _2\}<0\) and \(\alpha _1 \ne \alpha _2\) if the singularities are located in two antipodal points.
Theorem 1.3
Assume \(h=e^{-4\pi \alpha _1 G_{p_1}-4\pi \alpha _2 G_{p_2}}\) with \(p_2= -p_1\), \(\alpha _1= \min \{\alpha _1,\alpha _2\}<0\) and \(\alpha _1\ne \alpha _2\); then \(\forall \; u\in H^1(S^2)\)
When \(\alpha _1=\alpha _2<0\) Theorem 1.1 cannot be directly applied because (4) has solutions. However, it is possible to use a stereographic projection and a classification result in [24] to find an explicit expression for the solutions. In particular a direct computation allows to prove that all the solutions are minima of \(J_{\overline{\rho }}\) and to find the value of \(\displaystyle {\min _{H^1(S^2)}J_{\overline{\rho }}}\).
Theorem 1.4
Assume \(h=e^{-4\pi \alpha \left( G_{p_1}+G_{p_2}\right) }\) with \(\alpha <0\) and \(p_1=-p_2\); then \(\forall \; u\in H^1(S^2)\) we have
Moreover the following conditions are equivalent:
-
\(u\) realizes equality.
-
If \(\pi \) denotes the stereographic projection from \(p_1\) then
$$\begin{aligned} u\circ \pi ^{-1}(y)= 2\log \left( \frac{(1+|y|^2)^{1+\alpha }}{1+e^\lambda |y|^{2(1+\alpha )}} \right) +c \end{aligned}$$for some \(\lambda ,c \in {\mathbb {R}}\).
-
\(h e^{u} g_0\) is a metric with constant positive Gaussian curvature and conical singularities of order \(\alpha _i\) in \(p_i\), \(i=1,2\).
This is a generalization of Onofri’s inequality (2) for metrics with two conical singularities.
2 Preliminaries and Blow-Up Analysis
Let \((\Sigma ,g)\) be a smooth compact, connected, Riemannian surface and let \(S:=\{p_1,\ldots ,p_m\}\) be a finite subset of \(\Sigma \). Let us consider a function \(h\) satisfying (5) with \(K\in C^\infty (\Sigma )\), \(K>0\) and \(\alpha _i\in (-1,+\infty )\backslash \{0\}\). In order to distinguish the singular points of \(h\) from the regular ones, we introduce a singularity index function
We will denote \(\displaystyle {\alpha := \min _{p\in \Sigma } \beta (p)= \min \left\{ \min _{1\le i\le m}\alpha _i,0\right\} }\) the minimum singularity order. We shall consider the functional
Our goal is to give a sharp version of (7) finding the explicit value of
To simplify the notation we will set \(\overline{\rho }:=8\pi (1+\alpha )\), \(\rho _\varepsilon = \overline{\rho }-\varepsilon \), \(J_\varepsilon := J_{\rho _\varepsilon }\) and \(J:= J_{\overline{\rho }}\). From (7) it follows that \(\forall \; \varepsilon >0\) the functional \(J_\varepsilon \) is coercive and, by direct methods, it is possible to find a function \(u_\varepsilon \in H^1(\Sigma )\) satisfying
and
Since \(J_{\varepsilon }\) is invariant under addition of constants \(\forall \; \varepsilon >0\), we may also assume
Remark 2.1
\(u_\varepsilon \in C^{0,\gamma }(\Sigma )\cap W^{1,s}(\Sigma )\) for some \(\gamma \in (0,1)\) and \(s>2\).
Proof
It is easy to see that \(h\in L^q(\Sigma )\) for some \(q >1\) ( \(q=+\infty \) if \(\alpha =0\) and \(q< -\frac{1}{\alpha }\) for \(\alpha <0)\). Applying locally Remarks 2 and 5 in [7] one can show that \(u_\varepsilon \in L^{\infty }(\Sigma )\) so \(-\Delta {u_\varepsilon }\in L^q(\Sigma )\) and by standard elliptic estimates \(u_\varepsilon \in W^{2,q}(\Sigma )\). Since \(q>1\) the conclusion follows by Sobolev’s embedding theorems.\(\square \)
The behavior of \(u_\varepsilon \) is described by the following concentration-compactness result:
Proposition 2.1
Let \(u_n\) be a sequence satisfying
and
where \(\Vert \psi _n\Vert _{L^s(\Sigma )}\le C_2\) for some \(s>1\), and
with \(K_n\in C^\infty (\Sigma )\), \(0<a\le K_n\le b\) and \(\alpha _i >-1\), \(i=1,\ldots ,m\). Then there exists a subsequence \(u_{n_k}\) of \(u_n\) such that the following alternatives hold:
-
1.
\(u_{n_k}\) is uniformly bounded in \(L^\infty (\Sigma )\);
-
2.
\(u_{n_k}\longrightarrow -\infty \) uniformly on \(\Sigma \);
-
3.
there exist a finite blow-up set \(B =\{q_1,\ldots ,q_l\}\subseteq \Sigma \) and a corresponding family of sequences \(\{q^j_k\}_{k\in {\mathbb {N}}}\), \(j=1,\ldots , l\) such that \(q_k^j\mathop {\longrightarrow }\limits ^{k\rightarrow \infty } q_j\) and \(u_{n_k}(q_k^j)\mathop {\longrightarrow }\limits ^{k\rightarrow \infty } +\infty \) \(j=1,\ldots ,l\). Moreover \(u_{n_k}\mathop {\longrightarrow }\limits ^{k\rightarrow \infty } -\infty \) uniformly on compact subsets of \(\Sigma \backslash B \) and \(V_{n_k}e^{u_{n_k}} \rightharpoonup \sum _{j=1}^l \beta _j \delta _{q_j}\) weakly in the sense of measures where \(\beta _j = 8\pi (1+\beta (q_j))\) for \(j=1,\ldots ,l\).
A proof of Proposition 2.1 in the regular case can be found in [19] while the general case is a consequence of the results in [1, 5]. In our analysis we will also need the following local version of Proposition 2.1 proved by Li and Shafrir [20]:
Proposition 2.2
Let \(\Omega \) be an open domain in \({\mathbb {R}}^2\) and \(v_n\) be a sequence satisfying \(\Vert e^{v_n}\Vert _{L^1(\Omega )}\le C\) and
where \(0\le V_n\in C_0(\overline{\Omega })\) and \(V_n\longrightarrow V\) uniformly in \(\overline{\Omega }\). If \(v_n\) is not uniformly bounded from above on compact subset of \(\Omega \), then \(V_n e^{v_n} \rightharpoonup \displaystyle {8\pi \sum _{j=1}^lm_j \delta _{q_j}}\) as measures, with \(q_j\in \Omega \) and \(m_j\in \mathbb {N}^+\), \(j=1,\ldots ,l\).
Applying Proposition 2.1 to \(u_\varepsilon \) under the additional condition (12) we obtain that either \(u_\varepsilon \) is uniformly bounded in \(L^\infty (\Sigma )\) or its blow-up set contains a single point \(p\) such that \(\beta (p)= \alpha \). In the first case, one can use elliptic estimates to find uniform bounds on \(u_\varepsilon \) in \(W^{2,q}(\Sigma )\), for some \(q>1\); consequently, a subsequence of \(u_\varepsilon \) converges in \(H^{1}(\Sigma )\) to a function \(u\in H^1(\Sigma )\) that is a minimum point of \(J\) and a solution of (4) for \(\rho =\overline{\rho }\). We now focus on the second case, that is
By Proposition 2.1 we also get:
Lemma 2.1
If \(u_\varepsilon \) satisfies (11), (12) and (13), then, up to subsequences,
-
1.
\(\rho _\varepsilon h e^{u_\varepsilon } \rightharpoonup \overline{\rho }\; \delta _p\);
-
2.
\(u_\varepsilon \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}-\infty \) uniformly in \(\Omega \), \(\forall \;\Omega \subset \subset \Sigma \backslash \{p\}\);
-
3.
\(\overline{u}_\varepsilon \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}-\infty \);
-
4.
There exist \(\gamma \in (0,1)\), \(s>2\) such that \(u_\varepsilon -\overline{u_\varepsilon } \mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0} \overline{\rho }\;G_p\) in \(C^{0,\gamma }(\overline{\Omega })\cap W^{1,s}(\Omega )\) \(\forall \) \(\Omega \subset \subset \Sigma \backslash \{p\}\);
-
5.
\(\nabla u_\varepsilon \) is bounded in \(L^q(\Sigma )\) \(\forall \; q\in (1,2)\).
Proof
1., 2. and 3. are direct consequences of Proposition 2.1. To prove 4., we consider the Green’s representation formula
We stress that the Green’s function has the following properties:
-
\(\displaystyle {|G_x(y)|\le C_1 (1+ |\log d(x,y)|)}\) \(\forall \; x,y \in \Sigma \), \(x\ne y\).
-
\(\displaystyle {|\nabla ^x_g G_x(y)|}\le \frac{C_2}{d(x,y)}\) \(\forall \; x,y\in \Sigma \), \(x \ne y\).
-
\(G_x(y)= G_y(x)\) \(\forall \; x,y\in \Sigma \), \(x\ne y\).
Take \(q>1\) such that \(h\in L^q(\Sigma )\). The first property also yields
Let us fix \(\delta >0\) such that \(B_{3\delta }(p)\subset \Sigma \backslash \Omega \) and take a cut-off function \(\varphi \) such that \(\varphi \equiv 1\) in \(B_{\delta }(p)\) and \(\varphi \equiv 0\) in \(\Sigma \backslash B_{2\delta }(p)\).
By (14) and 2. we have
By 1. and the smoothness of \(\varphi G_x\) for \(x\in \overline{ \Omega }\) and \(y\in \Sigma \) we get
uniformly for \(x \in \Omega \). Similarly we have
with
uniformly in \(\Omega \) and, assuming \(q\in (1,2)\), by the Hardy–Littlewood–Sobolev inequality
where
Note that \(q>1\) implies \(s>2\). Finally, to prove 5., we shall observe that for any \(1<q<2\) there exists a positive constant \(C_q\) such that
Hence \(\forall \; \varphi \in W^{1,q'}(\Sigma )\)
so that
\(\square \)
We now focus on the behavior of \(u_\varepsilon \) near the blow-up point. First we consider the case \(\alpha <0\). Let us fix a system of normal coordinates in a small ball \(B_\delta (p)\), with \(p\) corresponding to \(0\) and \(p_\varepsilon \) corresponding to \(x_\varepsilon \). We define
Lemma 2.2
If \(\alpha <0\), \(\displaystyle { \frac{|x_\varepsilon |}{t_\varepsilon }} \) is bounded.
Proof
We define
where \(s_\varepsilon (x)\) is the solution of
The function \(\psi _\varepsilon \) satisfies
in \(B_\frac{\delta }{|x_\varepsilon |}(0)\). We stress that, by standard elliptic estimates, \(s_\varepsilon \) is uniformly bounded in \(C^1(\overline{B_\delta })\) and that \(G_p\) has the expansion
in \(B_\delta (0)\). Thus
where \(\tilde{h}\in C^{1}(\overline{B_{\delta }})\). In particular \(V_\varepsilon \) is uniformly bounded in \(C^1_{loc}({\mathbb {R}}^2\backslash \{0\})\). If there existed a subsequence such that \(\displaystyle { \frac{|x_\varepsilon |}{t_\varepsilon }}\longrightarrow +\infty \) then
so \(\displaystyle {y_0:= \lim _{\varepsilon \rightarrow 0} \frac{x_\varepsilon }{|x_\varepsilon |}}\) would be a blow-up point for \(\psi _\varepsilon \). Since \(y_0\ne 0\), applying Proposition 2.2 to \(\psi _\varepsilon \) in a small ball \(B_r(y_0)\) we would get
But this would be in contradiction to (12) since
\(\square \)
Lemma 2.3
Assume \(\alpha <0\). Then, possibly passing to a subsequence, \(\varphi _\varepsilon \) converges uniformly on compact subsets of \({\mathbb {R}}^2\) and in \(H^1_{loc}({\mathbb {R}}^2)\) to
where \(\displaystyle {c(p)= K(p)e^{-4\pi \alpha A(p)} \prod _{1\le i\le m, p_i\ne p} e^{-4\pi \alpha _i G_{p_i}(p)}}\).
Proof
The function \(\varphi _\varepsilon \) is defined in \(B_\varepsilon =B_\frac{\delta }{t_\varepsilon }(0)\) and satisfies
and
As in the previous proof we have
in \(L^q_{loc}({\mathbb {R}}^2)\) for some \(q>1\). Fix \(R>0\) and let \(\psi _\varepsilon \) be the solution of
Since \(\Delta \psi _\varepsilon \) is bounded in \(L^q(B_R(0))\) with \(q>1\), elliptic regularity shows that \(\psi _\varepsilon \) is bounded in \(W^{2,q}(B_R(0))\) and by Sobolev’s embeddings we may extract a subsequence such that \(\psi _\varepsilon \) converges in \(H^1(B_R(0))\cap C^{0,\lambda }(B_R(0))\). The function \(\xi _\varepsilon = \varphi _\varepsilon -\psi _\varepsilon \) is harmonic in \(B_R\) and bounded from above. Furthermore \(\xi _\varepsilon \left( \frac{x_\varepsilon }{t_\varepsilon }\right) = -\psi _\varepsilon \left( \frac{x_\varepsilon }{t_\varepsilon }\right) \) is bounded from below, hence by Harnack inequality \(\xi _\varepsilon \) is uniformly bounded in \(C^{2}(\overline{B_{\frac{R}{2}}}(0))\). Thus \(\varphi _\varepsilon \) is bounded in \(W^{2,q}(B_{\frac{R}{2}})\) and we can extract a subsequence converging in \(H^1(B_{\frac{R}{2}})\cap C^{0,\lambda }(B_\frac{R}{2})\). Using a diagonal argument we find a subsequence for which \(\varphi _\varepsilon \) converges in \(H^1_{loc}({\mathbb {R}}^2)\cap C^{0,\lambda }_{loc}({\mathbb {R}}^2)\) to a function \(\varphi _0\) solving
on \({\mathbb {R}}^2\) with
The classification result in [24] yields
for some \(\lambda \in {\mathbb {R}}\). To conclude the proof it remains to note that, since \(0\) is the unique maximum point of \(\varphi _0\), the uniform convergence of \(\varphi _\varepsilon \) implies \(\frac{x_\varepsilon }{t_\varepsilon }\longrightarrow 0\) and \(\lambda =0\).\(\square \)
As in [15], to give a lower bound on \(J_\varepsilon (u_\varepsilon )\) we need the following estimate from below for \(u_\varepsilon \):
Lemma 2.4
Fix \(R>0\) and define \(r_\varepsilon = t_\varepsilon R\). If \(\alpha <0\) and \(u_\varepsilon \) satisfies (11), (12), (13), then
in \( \Sigma \backslash B_{r_\varepsilon }(p)\), where \(o_\varepsilon (1)\) is a function of \(\varepsilon \) and \(R\) such that \(o_\varepsilon (1)\longrightarrow 0\) as \(\varepsilon \rightarrow 0\).
Proof
\(\forall \; C>0\) we have
Let us consider normal coordinates near \(p\). We know that
so by Lemma 2.3 if \(x=t_\varepsilon y\) with \(|y| =R\) we have
Thus, taking
we have \(u_\varepsilon -\overline{\rho } G_p -C_{\varepsilon ,R}\ge 0\) on \(\partial B_{r_\varepsilon }(p)\) and the conclusion follows from the maximum principle.\(\square \)
As a consequence we also have
Lemma 2.5
If \(u_\varepsilon \) and \(t_\varepsilon \) are as above, then \(t_\varepsilon ^2 \overline{u}_\varepsilon \longrightarrow 0\).
Proof
By Lemma 2.3
and by the previous lemma
Thus \(\dfrac{|\overline{u}_\varepsilon |}{\lambda _\varepsilon }\) is bounded and, since \(\lambda _\varepsilon t_\varepsilon ^2 =o_\varepsilon (1)\), we get the conclusion.\(\square \)
The case \(\alpha =0\) can be studied in a similar way. The main difference is that, since we do not know whether \(\frac{|x_\varepsilon |}{t_\varepsilon }\) is bounded, we have to center the scaling in \(p_\varepsilon \) and not in \(p\). Note that \(\beta (p)=0\) means that \(p\in \Sigma \backslash S\) is a regular point of \(h\).
Lemma 2.6
Assume that \(\alpha =0\) and that \(u_\varepsilon \) satisfies (11), (12) and (13). In normal coordinates near \(p\) define
Then
-
1.
\(\psi _\varepsilon \) converges in \(C^1_{loc}({\mathbb {R}}^2)\) to
$$\begin{aligned} \psi _0(x)= -2\log (1+\pi h(p) |x|^2) \end{aligned}$$ -
2.
\(\forall \;R>0\) one has
$$\begin{aligned} u_\varepsilon \ge 8\pi G_{p_\varepsilon } - \lambda _\varepsilon -8\pi A(p) + 2\log \left( \frac{R^2}{1+\pi h(p) R^2}\right) +o_\varepsilon (1) \end{aligned}$$in \(\Sigma \backslash B_{R t_\varepsilon }(p_\varepsilon );\)
-
3.
\(t_\varepsilon ^2 \overline{u}_\varepsilon \rightarrow 0\).
3 A Lower Bound
In this section and in the next one we present the proof of Theorem 1.1. We begin by giving an estimate from below of \(\displaystyle {\inf _{H^1(\Sigma )}J}\). As before we consider \(u_\varepsilon \) satisfying (10), (11), (12), and (13). Again we will focus on the case \(\alpha <0\) since the computation for \(\alpha =0\) is equivalent to the one in [15]. We consider normal coordinates in a small ball \(B_\delta (p)\) and assume that \(G_p\) has the expansion (16) in \(B_\delta (p)\). Let \(t_\varepsilon \) be defined as in (15), then \(\forall \;R>0\) we shall consider the decomposition
Throughout this section, \(o_\delta (1)\) (and \(o_R(1)\)) will denote a function depending only on \(\delta \) (resp. \(R\)) which converges to 0 as \(\delta \rightarrow 0\) (resp. \(R\rightarrow \infty \)), while the notation \(o_\varepsilon (1)\) will be used for functions of \(\varepsilon , \delta \) and \(R\) such that, for fixed \(\delta \) and \(R\), \(o_\varepsilon (1) \longrightarrow 0\) as \(\varepsilon \rightarrow 0\).
On \(\Sigma \backslash B_\delta (p)\) we can use Lemma 2.1 and an integration by parts to obtain:
On \(B_{r_\varepsilon }(p)\) the convergence result for the scaling (15) stated in Lemma 2.3 yields
For the remaining term we can use (11) and Lemma 2.1 to obtain
and
Using (19), (20), (21) and (22) we get
By Lemmas 2.1 and 2.5 we can say that
Using Green’s formula we find
Similarly
and
Lemma 2.3 yields
and the estimate in Lemma 2.4 gives
Hence
By (17), (18) and (24) we can therefore conclude
so that
As \(\varepsilon ,\delta \rightarrow 0\) and \(R\rightarrow \infty \) we obtain
Using Lemma 2.6 it is possible to prove that (25) holds even for \(\alpha =0\). About the blow-up point \(p\) we only know that \(\beta (p)=\alpha \), so we have proved
Proposition 3.1
If \(J\) has no minimum point, then
Notice that, if \(\alpha <0\), the set
is finite, while if \(\alpha =0\)
Although this set is not finite, the maximum in the above expression is still well defined since the function
is continuous on \(\Sigma \backslash S\) and approaches \(-\infty \) near \(S\).
4 An Estimate from Above
In order to complete the proof of Theorem 1.1 we need to exhibit a sequence \(\varphi _\varepsilon \in H^1(\Sigma )\) such that
Let us define \(r_\varepsilon :=\gamma _\varepsilon \varepsilon ^\frac{1}{2(1+\alpha )}\) where \(\gamma _\varepsilon \) is chosen so that
Let \(p\in \Sigma \) be such that \(\beta (p)=\alpha \) and
and consider a cut-off function \(\eta _\varepsilon \) such that \(\eta _\varepsilon \equiv 1\) in \(B_{r_\varepsilon }(p)\), \(\eta _\varepsilon \equiv 0\) in \(\Sigma \backslash B_{2 r_\varepsilon }(p)\) and \(|\nabla _g \eta _\varepsilon |=O(r_\varepsilon ^{-1})\). Define
where \(r=d(x,p)\), \(\sigma (x)=O(r)\) is defined by
and
In the case \(\alpha _i=0\) \(\forall \;i\), a similar family of functions was used in [15] to give an existence result for (4) by proving, under some strict assumptions on \(h\), that
Here we only prove a weak inequality but we have no extra assumptions on \(h\). Taking normal coordinates in a neighborhood of \(p\) it is simple to verify that
By our definition of \(\varphi _\varepsilon \)
and by the properties of \(\eta _\varepsilon \)
Hence, integrating by parts and using (27), one has
Thus
Similarly one has
and
so that
To compute the integral of the exponential term we fix a small \(\delta >0\) and observe that
where \(\displaystyle {\tilde{h} = h\; e^{4\pi \alpha G_p}= K \prod _{q\in S,q\ne p}}e^{-4\pi \beta (q) G_q}\). For the first term we have
Since \(\tilde{h}\) is smooth in a neighborhood of \(p\) we obtain
and
Finally
so by (30), (31), (32) and (33) we have
Using (28), (29) and (34) we get
This, together with Proposition 3.1, completes the proof of Theorem 1.1.
5 Onofri’s Inequalities on \(S^2\)
In this section we will consider the special case of the standard sphere \((S^2,g_0)\) with \(m\le 2\) and \(K\equiv 1\). We fix \(\alpha _1,\alpha _2\in {\mathbb {R}}\) with \(-1<\alpha _1\le \alpha _2\) and as before we consider the singular weight
In order to apply Theorem 1.1 and obtain sharp versions of (7), we need to study the existence of minimum points for the functional \(J\). Let us fix a system of coordinates \((x_1,x_2,x_3)\) on \({\mathbb {R}}^3\) such that \(p_1=(0,0,1)\). When \(h\in C^1(S^2)\), the Kazdan–Warner identity (see [18]) states that any solution of (4) has to satisfy
We claim that if \(p_2=-p_1\) the same identity holds, at least in the \(x_3\)-direction, even when \(h\) is singular.
Lemma 5.1
Let \(u\) be a solution of (4) on \(S^2\), then there exist \(C,\delta _0>0\) such that
for \(0<d(x,p_i)<\delta _0\), \(\;i=1,2\).
Proof
Let us fix \(0<r_0<\frac{1}{2}\min \{\frac{\pi }{2},d(p_1,p_2)\}\) and \(i\in \{1,2\}\). If \(\alpha _i>- \frac{1}{2}\) then, by standard elliptic regularity, \(u \in C^1(\overline{B_{r_0}(p_i)})\) and the conclusion holds for \(\delta _0=r_0\) and \(C=\Vert \nabla u\Vert _{L^\infty (B_{r_0}(p_i))}\). Let us now assume \(\alpha _i \le -\frac{1}{2}\). We know that \(h(y)\le C_1 d(y,p_i)^{2\alpha _i}\) for \(y\in B_{2r_0}(p_i)\) so, if \(\delta _0 < r_0\), by Green’s representation formula we have
Let \(\pi \) be the stereographic projection from the point \(-p_i\). It is easy to check that there exist \(C_2,C_3>0\) such that
\(\forall \; q,q'\in B_{\frac{\pi }{2}}(p_i)\). Thus we have
Notice that
If \(\alpha _i<-\frac{1}{2}\)
while if \(\alpha _i=-\frac{1}{2}\)
Thus we get the conclusion for \(\delta _0\) sufficiently small.\(\square \)
In any case there exists \(s\in [0,1)\) such that
for \(0<d(x,p_i)<\delta _0\), \(\;i=1,2\).
Proposition 5.1
If \(p_2=-p_1\) then any solution of (4) satisfies
Proof
Without loss of generality we may assume
Let us denote \(S_\delta = S^2\backslash B_{\delta }(p_1)\cup B_\delta (p_2)\). Since \(u\) is smooth in \(S_\delta \), multiplying (4) by \(\nabla u\cdot \nabla x_3\) and integrating on \(S_\delta \) we have
Integrating by parts we obtain
and by (35)
Using the identities
and
and applying again (35) to estimate the boundary term, we get
Thus (37) becomes
Moreover
and
Thus by (38) we have
Since \(u\) is continuous on \(S^2\) and \(h, \nabla h \cdot \nabla x_3\in L^1(S^2)\) as \(\delta \rightarrow 0 \) we get the conclusion.\(\square \)
Remark 5.1
In this proof there is no need to assume \(K\equiv 1\).
Assuming \(p_1=(0,0,1)\) and \(p_2=(0,0,-1)\), one may easily verify that
and
so that
Thus we can rewrite the identity in Proposition 5.1 as
Proof of Theorem 1.2
Assume \(m=1\) (i.e., \(\alpha _2=0\)). We claim that equation (4) has no solutions for \(\rho = \overline{\rho } = 8\pi (1+\min \{0,\alpha _1\})\), unless \(\alpha _1 =0\). Indeed if \(u\) were a solution of (4) satisfying (36), then applying (39) with \(\rho = \overline{\rho }\) we would get
so that, if \(\alpha _1\ne 0\),
This contradicts (4). In particular we proved non-existence of minimum points for \(J_{\overline{\rho }}\) so we can exploit Theorem 1.1 and (9) to prove that (7) holds with
If \(\alpha _1<0\) one has
If \(\alpha _1>0\),
\(\square \)
Proof of Theorem 1.3
As in the previous proof, applying (39) with \(\rho = \overline{\rho } =8\pi (1+\alpha _1)\), we obtain that any critical point of (4) for which (36) holds has to satisfy
Since \(\alpha _1 \ne \alpha _2\) one has
which is impossible. Thus \(J_{\overline{\rho }}\) has no critical points and by Theorem 1.1 one has
\(\square \)
Now we assume \(\alpha _1=\alpha _2<0\). In this case identity (39) gives no useful condition. Let us denote by \(\pi \) the stereographic projection from the point \(p_1\). It is easy to verify that \(u\) satisfies (4) and (36) if and only if
solves
in \({\mathbb {R}}^2\) and
As we pointed out in the proof of Lemma 2.3 and Eq. (40) has a one-parameter family of solutions:
\(l\in {\mathbb {R}}\). Thus we have a corresponding family \(\{u_{\lambda ,c}\}\) of critical points of \(J_{\overline{\rho }}\) given by the expression
\(c\in {\mathbb {R}},\lambda >0\). A priori we do not know whether these critical points are minima for \(J_{\overline{\rho }}\) (as it happens for \(\alpha =0\)), so a direct application of 1.1 is not possible. However, we can still get the conclusion by comparing \(J_{\overline{\rho }}(u_{\lambda ,c})\) with the blow-up value provided by Theorem 1.1.
Proof of Theorem 1.4
Let us first compute \(J(u_{\lambda ,c})\). Let \(\varphi _t:S^2\longrightarrow S^2\) be the conformal transformation defined by \(\pi (\varphi _t (\pi ^{-1}(y)))= t y\). It is not difficult to prove that \(\forall \; t>0\)
in particular, since
we have that \(J(u_{\lambda ,c})\) does not depend on \(\lambda \) and \(c\). Thus we may assume \(\lambda =1\) and \(c=0\). A simple computation shows that
Since \(u_{1,0}(p_1)=0\) and \(u_{1,0}\) solves
one has
and
Since
we get
Using (42), (43) and (44) we obtain
To conclude the proof it is sufficient to observe that \(u_{\lambda ,c}\) have to be minimum points for \(J_{\overline{\rho }}\) that is
Indeed if this were false then \(J_{\overline{\rho }}\) would have no minimum points but, by Theorem 1.1, we would get
This is clearly a contradiction.\(\square \)
Remark 5.2
There is no need to assume \(p_1=-p_2\).
Indeed given two arbitrary points \(p_1,p_2\in S^2\) with \(p_1\ne p_2\) it is always possible to find a conformal diffeomorphism \(\varphi :S^2 \longrightarrow S^2\) such that \(\varphi ^{-1}(p_1) = -\varphi ^{-1}(p_2)\). Moreover one has
\(\forall \; u \in H^1(S^2)\), where \(\widetilde{J}\) is the Moser–Trudinger functional associated to
and \(c_{\alpha ,p_1,p_2}\) is an explicitly known constant depending only on \(\alpha \), \(p_1\) and \(p_2\). In particular one can still compute \(\min _{H^1(S^2)} J_{\overline{\rho }}\) and describe the minimum points of \(J_{\overline{\rho }}\) in terms of \(\varphi \) and the family (41).
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Acknowledgments
The author would like to express his gratitude to Professor Andrea Malchiodi for many valuable discussions and for his guidance during the preparation of this work. The author is supported by the FIRB Project Analysis and Beyond, by the PRINs Variational Methods and Nonlinear PDE’s and Variational and perturbative aspects of nonlinear differential problems and by the Mathematics Department at the University of Warwick.
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Mancini, G. Onofri-Type Inequalities for Singular Liouville Equations. J Geom Anal 26, 1202–1230 (2016). https://doi.org/10.1007/s12220-015-9589-3
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DOI: https://doi.org/10.1007/s12220-015-9589-3