1 Introduction

An essential tool in the study of the embeddings of Sobolev spaces is the Moser–Trudinger inequality, which gives compact embedding in any \(L^p\) space for finite \(p\ge 1\) and also exponential integrability.

If we consider a 2-dimensional compact Riemannian manifold \((\Sigma ,g)\), due to well-known works from Moser [18] and Fontana [13] we get

(1)

where \(\nabla =\nabla _g\) is the gradient given by the metric g and \(C=C_{\Sigma ,g}\) is a constant depending only on \(\Sigma \) and g.

This inequality has fundamental importance in the study of the Liouville equations of the kind

$$\begin{aligned} -\Delta u=\rho \left(\frac{he^u}{\int _\Sigma he^u{\mathrm {d}}V_g}-1\right), \end{aligned}$$
(2)

where \(\Delta =\Delta _g\) is the Laplace-Beltrami operator, \(\rho \) a positive real parameter, h a positive smooth function and \(\Sigma \) is supposed, without loss of generality, to have area equal to \(|\Sigma |=1\). In fact, the solutions of (2) are critical points of the functional

$$\begin{aligned} I_\rho (u)=\frac{1}{2}\int _\Sigma |\nabla u|^2{\mathrm {d}}V_g-\rho \left(\log \int _\Sigma he^u{\mathrm {d}}V_g-\int _\Sigma u{\mathrm {d}}V_g\right); \end{aligned}$$

using the inequality (1) we can control the last term by the Dirichlet energy, thus showing that \(I_\rho \) is bounded from below on \(H^1(\Sigma )\) if and only if \(\rho \) is smaller or equal to \(8\pi \).

Equations like (2) have great importance in different contexts like the Gaussian curvature prescription problem (see for instance [6, 7]) and abelian Chern–Simons models in theoretical physics ([21, 24]).

An extension of the inequality (1), which takes into consideration power-type weights, was given by Chen [8] and Trojanov [22]. For a given \(p\in \Sigma \) and \(\alpha \in (-1,0]\), they showed that

$$\begin{aligned} (1+\alpha )\left(\log \int _\Sigma d(\cdot ,p)^{2\alpha }e^u{\mathrm {d}}V_g-\int _\Sigma u{\mathrm {d}}V_g\right)\le \frac{1}{16\pi }\int _\Sigma |\nabla u|^2{\mathrm {d}}V_g+C\quad \quad \quad \forall \,u\in H^1(\Sigma ). \end{aligned}$$
(3)

This inequality allows to treat singularities in the Eq. (2), that is to consider equations like

$$\begin{aligned} -\Delta u=\rho \left(\frac{he^u}{\int _\Sigma he^u{\mathrm {d}}V_g}-1\right)-4\pi \sum _{m=1}^M\alpha _m(\delta _{p_m}-1), \end{aligned}$$
(4)

where we take arbitrary \(p_1,\dots ,p_M\in \Sigma \) and \(\alpha _m>-1\) for any \(m\in \{1,\dots ,M\}\).

This is a natural extension of (2), which allows to consider the same problems in a more general context. For instance, it arises in the Gaussian curvature prescription problem on surfaces with conical singularities and in Chern–Simons vortices theory.

Defining \(G_p\) as the Green function of \(-\Delta \) on \(\Sigma \) centered at a point p, through the change of variables

$$\begin{aligned} u\mapsto u+4\pi \sum _{m=1}^M\alpha _mG_{p_m} \end{aligned}$$
(5)

Equation (4) becomes

$$\begin{aligned} -\Delta u=\rho \left(\frac{\widetilde{h}e^u}{\int _\Sigma \widetilde{h}e^u{\mathrm {d}}V_g}-1\right) \end{aligned}$$

with \(\widetilde{h}=he^{-4\pi \sum _{m=1}^M\alpha _mG_{p_m}}\).

Since \(G_p\) has the same behavior as \(\frac{1}{2\pi }\log \frac{1}{d(\cdot ,p)}\) around p, then \(\widetilde{h}\) behaves like \(d(\cdot ,p_m)^{2\alpha _m}\) around each singular point \(p_m\), hence the energy functional

$$\begin{aligned} I_\rho (u)=\frac{1}{2}\int _\Sigma |\nabla u|^2{\mathrm {d}}V_g-\rho \left(\log \int _\Sigma \widetilde{h}e^u{\mathrm {d}}V_g-\int _\Sigma u{\mathrm {d}}V_g\right) \end{aligned}$$

can be estimated, as in the regular case, using (3).

The purpose of this paper is to extend inequality (3) to singular Liouville systems of the type

$$\begin{aligned} -\Delta u_i=\sum _{j=1}^Na_{ij}\rho _j\left(\frac{h_je^{u_j}}{\int _\Sigma h_je^{u_j}{\mathrm {d}}V_g}-1\right)-4\pi \sum _{m=1}^M\alpha _{im}(\delta _{p_m}-1),\quad i=1,\dots ,N, \end{aligned}$$

where \(A=(a_{ij})\) is a \(N\times N\) symmetric positive definite matrix and \(\rho _i,h_i,\alpha _{im}\) are as before.

Applying, similarly to (5), the change of variables

$$\begin{aligned} u_i\mapsto u_i+4\pi \sum _{m=1}^M\alpha _{im}G_{p_m}, \end{aligned}$$

the system becomes

$$\begin{aligned} -\Delta u_i=\sum _{j=1}^Na_{ij}\rho _j\left(\frac{\widetilde{h}_je^{u_j}}{\int _\Sigma \widetilde{h}_je^{u_j}{\mathrm {d}}V_g}-1\right),\quad i=1,\dots ,N, \end{aligned}$$
(6)

with \(\widetilde{h}_j\) having the same behavior around the singular points.

The system has a variational formulation with the energy functional

$$\begin{aligned} J_\rho (u):=\frac{1}{2}\sum _{i,j=1}^Na^{ij}\int _\Sigma \nabla u_i\cdot \nabla u_j{\mathrm {d}}V_g-\sum _{i=1}^N\rho _i\left(\log \int _\Sigma \widetilde{h}_ie^{u_i}{\mathrm {d}}V_g-\int _\Sigma u_i{\mathrm {d}}V_g\right), \end{aligned}$$
(7)

with \(a^{ij}\) indicating the entries of the inverse matrix \(A^{-1}\) of A.

A recent paper by the author and Malchiodi ([2]) gives an answer for the particular case of the SU(3) Toda system, that is \(N=2\) and A is the Cartan matrix

$$\begin{aligned} \left(\begin{array}{c@{\quad }c} 2&{}-1\\ -1&{}2 \end{array} \right). \end{aligned}$$

This is a particularly interesting case, due to its application in the description of holomorphic curves in \(\mathbb {C}\mathbb {P}^N\) in geometry ([3, 5, 9]) and in the non-abelian Chern–Simons theory in physics ([12, 21, 24]).

The authors prove a sharp inequality, that is they show that the functional \(J_\rho \) is bounded from below if and only if both the parameters \(\rho _i\) are less or equal than \(4\pi \min \left\{ 1,1+\min _m\alpha _{im}\right\} \), thus extending the result in the regular case from [15].

Concerning general regular Liouville systems, Wang [23] gave necessary and sufficient conditions for the boundedness from below of \(J_\rho \), following previous results in [10, 11] for the problem on Euclidean domains with Dirichlet boundary conditions. Analogous results were given in [20] for the standard unit sphere \(\left(\mathbb {S}^2,g_0\right)\) and in [19] for a similar problem.

In these papers, the authors introduce, for any \(\mathcal { I}\subset \{1,\dots ,N\}\), the following function of the parameter \(\rho \):

$$\begin{aligned} \Lambda _{{\mathcal {I}}}(\rho )=8\pi \sum _{i\in {\mathcal {I}}}\rho _i-\sum _{i,j\in {\mathcal {I}}}a_{ij}\rho _i\rho _j. \end{aligned}$$

What they prove is boundedness from below for \(J_\rho \) for any \(\rho \in \mathbb {R}_+^N\) which satisfies \(\Lambda _{{\mathcal {I}}}(\rho )>0\) for all the subsets \({\mathcal {I}}\) of \(\{1,\dots ,N\}\), whereas \(\inf _{H^1(\Sigma )^N}J_\rho =-\infty \) whenever \(\Lambda _{{\mathcal {I}}}(\rho )<0\) for some \({\mathcal {I}}\subset \{1,\dots ,N\}\).

The first main result of this paper is an extension of the results from [10, 11, 23] to the case of singularities.

Similarly to Liouville equation, we will have to multiply some quantities by \(1+\alpha _{im}\). Precisely, we have:

Theorem 1.1

Let \(J_\rho \) be the functional defined by (7) and set, for \(\rho \in \mathbb {R}_{>0}^N,\,x\in \Sigma \) and \(i\in {\mathcal {I}}\subset \{1,\dots ,N\}\):

$$\begin{aligned} \alpha _i(x)= & {} \left\{ \begin{array}{ll}\alpha _{im}&{}\text{ if } x=p_m\\ 0&{}\text{ otherwise }\end{array}\right. \quad \quad \quad \Lambda _{{\mathcal {I}},x}(\rho ):=8\pi \sum _{i\in {\mathcal {I}}}(1+\alpha _i(x))\rho _i-\sum _{i,j\in {\mathcal {I}}}a_{ij}\rho _i\rho _j\nonumber \\ \Lambda (\rho ):= & {} \min _{{\mathcal {I}}\subset \{1,\dots ,N\},x\in \Sigma }\Lambda _{{\mathcal {I}},x}(\rho ). \end{aligned}$$
(8)

Then, \(J_\rho \) is bounded from below if \(\Lambda (\rho )>0\), whereas \(J_\rho \) is unbounded from below if \(\Lambda (\rho )<0\).

Notice that, in the definition of \(\Lambda \), the minimum makes sense because it is taken in a finite set, since \(\alpha _i(x)=0\) for all points of \(\Sigma \) but a finite number, and for all the former points \(\Lambda _{{\mathcal {I}},x}\) is defined in the same way.

As a consequence of this result, we obtain information about the existence of solutions for the system (6).

Corollary 1.2

The functional \(J_\rho \) is coercive in \(\overline{H}^1(\Sigma )\) if and only if \(\Lambda (\rho )>0\).

Therefore, if this occurs, then \(J_\rho \) admits a minimizer u which solves (6).

Theorem 1.1 leaves an open question about what happens when \(\Lambda (\rho )=0\). In this case, as we will see in the following Sections, one encounters blow-up phenomena which are not yet fully known for general systems.

Anyway, we can say something more if we assume in addition \(a_{ij}\le 0\) for any \(i,j\in \{1,\dots ,N\}\) with \(i\ne j\). First of all, we notice that in this case

$$\begin{aligned} \Lambda (\rho )= & {} \min _{i\in \{1,\dots ,N\}}\left(8\pi (1+\widetilde{\alpha }_i)\rho _i-a_{ii}\rho _i^2\right),\quad \quad \quad \text{ where }\nonumber \\ \widetilde{\alpha }_i:= & {} \min _{m\in \{1,\dots ,M\},x\in \Sigma }\alpha _i(x)=\min \left\{ 0,\min _{m\in \{1,\dots ,M\}}\alpha _{im}\right\} ; \end{aligned}$$
(9)

hence the sufficient condition in Theorem 1.1 is equivalent to assuming \(\rho _i<\frac{8\pi (1+\widetilde{\alpha }_i)}{a_{ii}}\) for any i.

With this assumption, studying what happens when \(\Lambda _{\mathcal I}(\rho )=0\) is reduced to a single-component local blow-up, which can be treated by using an inequality from [1]. Therefore, we get the following sharp result:

Theorem 1.3

Let \(J_\rho \) be defined by (7), \(\widetilde{\alpha }_i\) as in (9) and \(\Lambda (\rho )\) as in Theorem 1.1, and suppose \(a_{ij}\le 0\) for any \(i,j\in \{1,\dots ,N\}\) with \(i\ne j\).

Then, \(J_\rho \) is bounded from below on \(H^1(\Sigma )^N\) if and only if \(\Lambda (\rho )\ge 0\), namely if and only if \(\rho _i\le \frac{8\pi (1+\widetilde{\alpha }_i)}{a_{ii}}\) for any \(i\in \{1,\dots ,N\}\).

We remark that the assuming A to be positive definite is necessary. If it is not, then \(J_\rho \) is unbounded from below for any \(\rho \).

In fact, suppose there exists \(v\in \mathbb {R}^N\) such that \(\sum _{i,j=1}^Na^{ij}v_iv_j\le -\theta |v|^2\) for some \(\theta >0\). Then, we consider the family of functions \(u^\lambda (x):=\lambda v\cdot x\); by Jensen’s inequality we get

We also notice that, with respect to the scalar case, in Theorem 1.1 and Corollary 1.2 the positive coefficients \(\alpha _{im}\)’s may affect the definition of \(\Lambda (\rho )\), hence the conditions for coercivity and boundedness from below of \(J_\rho \).

On the other hand, under the assumptions of Theorem 1.3, coercivity and boundedness from below only depend on the negative \(\alpha _{im}\)’s, just like for the scalar equation.

The plan of this paper is the following: in Sect. 2 we will introduce some notations and some preliminary results which will be used throughout the rest of the paper. In Sect. 3 we will show a sort of Concentration-compactness theorem, showing the possible non-compactness phenomena for solutions of the system (6). Finally, in Sects. 4 and 5 we will give the proof of the two main theorems.

2 Notations and preliminaries

In this section, we will give some useful notation and some known preliminary results which will be needed to prove the two main theorems.

Given two points \(x,y\in \Sigma \), we will indicate the metric distance on \(\Sigma \) between them as d(x, y). We will indicate the open metric ball centered in p having radius r as

$$\begin{aligned} B_r(x):=\{y\in \Sigma :\,d(x,y)<r\}. \end{aligned}$$

For any subset of a topological space \(A\subset X\) we indicate its closure as \(\overline{A}\) and its interior part as \(\mathring{A}\).

Given a function \(u\in L^1(\Sigma )\), the symbol \(\overline{u}\) will indicate the average of u on \(\Sigma \). Since we assume \(|\Sigma |=1\), we can write:

We will indicate the subset of \(H^1(\Sigma )\) which contains the functions with zero average as

$$\begin{aligned} \overline{H}^1(\Sigma ):=\left\{ u\in H^1(\Sigma ):\,\overline{u}=0\right\} . \end{aligned}$$

Since the functional \(J_\rho \) defined by (7) is invariant by addition of constants, it will not be restrictive to study it on \(\overline{H}^1(\Sigma )^N\) rather than on \(H^1(\Sigma )^N\).

We will indicate with the letter C large constants which can vary among different lines and formulas. To underline the dependence of C on some parameter \(\alpha \), we indicate with \(C_\alpha \) and so on.

We will denote as \(o_\alpha (1)\) quantities which tend to 0 as \(\alpha \) tends to 0 or to \(+\infty \) and we will similarly indicate bounded quantities as \(O_\alpha (1)\), omitting in both cases the subscript(s) when it is evident from the context.

First of all, we need a result from Brezis and Merle [4]. It is a classical estimate about exponential integrability of solutions of some elliptic PDEs.

Lemma 2.1

([4], Theorem 1) Take \(r>0,\,\Omega :=B_r(0)\subset \mathbb {R}^2,\,f\in L^1(\Omega )\) with \(\Vert f\Vert _{L^1(\Omega )}<4\pi \) and u solving

$$\begin{aligned} \left\{ \begin{array}{ll}-\Delta u=f&{}\text{ in } \Omega \\ u=0&{}\text{ on } \partial \Omega \end{array}\right. . \end{aligned}$$

Then, for any \(q\in \left[1,\frac{4\pi }{\Vert f\Vert _{L^1(\Omega )}}\right)\) there exists a constant \(C=C_{q,r}\) such that \(\int _\Omega e^{q|u(x)|}{\mathrm {d}}x\le C\).

A crucial role in the proof of both Theorems 1.1 and 1.3 will be played by the concentration values of the sequences of solutions of (6).

For a sequence \(u^n=\left\{ u_1^n,\dots ,u_N^n\right\} _{n\in \mathbb {N}}\) of solutions of (6) with \(\rho =\rho ^n=\left\{ \rho _1^n,\dots ,\rho _N^n\right\} \), we define (up to subsequences), for \(i\in \{1,\dots ,N\}\), the concentration value of its \(i^{th}\) component around a point \(x\in \Sigma \) as

$$\begin{aligned} \sigma _i(x):=\lim _{r\rightarrow 0}\lim _{n\rightarrow +\infty }\rho _i^n\frac{\int _{B_r(x)}\widetilde{h}_ie^{u_i^n}{\mathrm {d}}V_g}{\int _\Sigma \widetilde{h}_ie^{u_i^n}{\mathrm {d}}V_g}. \end{aligned}$$
(10)

In a recent paper ([16], see also [14] for the regular case) it was proved, by a Pohožaev identity, that the concentration values satisfy the following algebraic relation, which involves the same quantities as in Theorem 1.1:

Proposition 2.2

([14], Lemma 2.2; [16], Proposition 3.1) Let \(\left\{ u^n\right\} _{n\in \mathbb {N}}\) be a sequence of solutions of (6), \(\alpha _i(x)\) and \(\Lambda _{{\mathcal {I}},x}\) as in (8) and \(\sigma (x)=(\sigma _1(x),\dots ,\sigma _N(x))\) as in (10). Then,

$$\begin{aligned} \Lambda _{\{1,\dots ,N\},x}(\sigma (x))=8\pi \sum _{i=1}^N(1+\alpha _i(x))\sigma _i(x)-\sum _{i,j=1}^Na_{ij}\sigma _i(x)\sigma _j(x)=0. \end{aligned}$$

To study the concentration phenomena of solutions of (6) we will use the following simple but useful calculus Lemma:

Lemma 2.3

([15], Lemma 4.4) Let \(\left\{ a^n\right\} _{n\in \mathbb {N}}\) and \(\left\{ b^n\right\} _{n\in \mathbb {N}}\) two sequences of real numbers satisfying

$$\begin{aligned} a^n\underset{n\rightarrow +\infty }{\longrightarrow }+\infty \quad \quad \quad \lim _{n\rightarrow +\infty }\frac{b^n}{a^n}\le 0. \end{aligned}$$

Then, there exists a smooth function \(F:[0,+\infty )\rightarrow \mathbb {R}\) which satisfies, up to subsequences,

$$\begin{aligned} 0<F'(t)<1\quad \forall \,t>0\quad \quad \quad F'(t)\underset{t\rightarrow +\infty }{\longrightarrow }0\quad \quad \quad F\left(a^n\right)-b^n\underset{n\rightarrow +\infty }{\longrightarrow }+\infty . \end{aligned}$$

Finally, as anticipated in the introduction, we will need a singular Moser–Trudinger inequality for Euclidean domains by Adimurthi and Sandeep [1], and its straightforward corollary.

Theorem 2.4

([1], Theorem 2.1) For any \(r>0,\,\alpha \in (-1,0]\) there exists a constant \(C=C_{\alpha ,r}\) such that if \(\Omega :=B_r(0)\subset \mathbb {R}^2\) and \(u\in H^1_0(\Omega )\), then

$$\begin{aligned} \int _\Omega |\nabla u(x)|^2{\mathrm {d}}x\le 1\Rightarrow \int _\Omega |x|^{2\alpha }e^{4\pi (1+\alpha )u(x)^2}{\mathrm {d}}x\le C \end{aligned}$$

Corollary 2.5

For any \(r>0,\,\alpha \in (-1,0]\) there exists a constant \(C=C_{\alpha ,r}\) such that if \(\Omega :=B_r(0)\subset \mathbb {R}^2\) and \(u\in H^1_0(\Omega )\), then

$$\begin{aligned} (1+\alpha )\log \int _\Omega |x|^{2\alpha }e^{u(x)}{\mathrm {d}}x\le \frac{1}{16\pi }\int _\Omega |\nabla u(x)|^2{\mathrm {d}}x+C \end{aligned}$$

Proof

By the elementary inequality \(u\le \theta u^2+\frac{1}{4\theta }\) with \(\theta =\frac{4\pi (1+\alpha )}{\int _\Omega |\nabla u(y)|^2{\mathrm {d}}y}\) we get

$$\begin{aligned} (1+\alpha )\log \int _\Omega |x|^{2\alpha }e^{u(x)}{\mathrm {d}}x\le & {} (1+\alpha )\log \int _\Omega |x|^{2\alpha }e^{\theta u(x)^2+\frac{1}{4\theta }}{\mathrm {d}}x\\= & {} \frac{1}{16\pi }\int _\Omega |\nabla u(y)|^2{\mathrm {d}}y+(1+\alpha )\log \\&\times \int _\Omega |x|^{2\alpha }e^{4\pi (1+\alpha )\left(\frac{u(x)}{\sqrt{\int _\Omega |\nabla u(y)|^2{\mathrm {d}}y}}\right)^2}{\mathrm {d}}x\\\le & {} \frac{1}{16\pi }\int _\Omega |\nabla u(y)|^2{\mathrm {d}}y+C. \end{aligned}$$

\(\square \)

3 A Concentration-compactness theorem

The aim of this section is to prove a result which describes the concentration phenomena for the solutions of (6), extending what was done for the two-dimensional Toda system in [2, 17].

We actually have to normalize such solutions to bypass the issue of the invariance by translation by constants and to have the parameter \(\rho \) multiplying only the constant term.

In fact, for any solution u of (6) the functions

$$\begin{aligned} v_i:=u_i-\log \int _\Sigma \widetilde{h}_ie^{u_i}{\mathrm {d}}V_g+\log \rho _i \end{aligned}$$
(11)

solve

$$\begin{aligned} \left\{ \begin{array}{l}-\Delta v_i=\sum _{j=1}^Na_{ij}\left(\widetilde{h}_je^{v_j}-\rho _j\right)\\ \int _\Sigma \widetilde{h}_ie^{v_i}{\mathrm {d}}V_g=\rho _i\end{array}\right. \quad \quad \quad i=1,\dots ,N. \end{aligned}$$
(12)

Moreover, we can rewrite in a shorter way (10) as

$$\begin{aligned} \sigma _i(x)=\lim _{r\rightarrow 0}\lim _{n\rightarrow +\infty }\int _{B_r(x)}\widetilde{h}_i^ne^{v_i^n}{\mathrm {d}}V_g. \end{aligned}$$

For such functions, we get the following concentration-compactness alternative:

Theorem 3.1

Let \(\{u^n\}_{n\in \mathbb {N}}\) be a sequence of solutions of (6) with \(\rho ^n\underset{n\rightarrow +\infty }{\longrightarrow }\rho \in \mathbb {R}_+^N\) and \(\widetilde{h}_i^n=V_i^n\widetilde{h}_i\) with \(V_i^n\underset{n\rightarrow +\infty }{\longrightarrow }1\) in \(C^1(\Sigma )^N\), \(\{v^n\}_{n\in \mathbb {N}}\) be defined as in (11) and \(\mathcal S_i\) be defined, for \(i\in \{1,\dots ,N\}\), by

$$\begin{aligned} \mathcal S_i:=\left\{ x\in \Sigma :\,\exists \,x^n\underset{n\rightarrow +\infty }{\longrightarrow }x \text{ such } \text{ that } v_i^n\left(x^n\right)\underset{n\rightarrow +\infty }{\longrightarrow }+\infty \right\} . \end{aligned}$$
(13)

Then, up to subsequences, one of the following occurs:

  • If \(\mathcal S_i=\emptyset \) for any \(i\in \{1,\dots ,N\}\), then \(v^n\underset{n\rightarrow +\infty }{\longrightarrow }v\) in \(W^{2,q}(\Sigma )^N\) for some \(q>1\) and some v which solves (12).

  • If \(\mathcal S_i\ne \emptyset \) for some i, then it is a finite set for all such i’s. If this occurs, then there is a subset \(\mathcal I\subset \{1,\dots ,N\}\) such that \(v_j^n\underset{n\rightarrow +\infty }{\longrightarrow }-\infty \) in \(L^\infty _{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j'=1}^N\mathcal S_{j'}\right)\) for any \(j\in {\mathcal {I}}\) and \(v_j^n\underset{n\rightarrow +\infty }{\longrightarrow }v_j\) in \(W^{2,q}_{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j'=1}^N\mathcal S_{j'}\right)\) for some \(q>1\) and some suitable \(v_j\), for any \(j\in \{1,\dots ,N\}{{\setminus }}\mathcal I\).

Since \(\widetilde{h}_j\) is smooth outside the points \(p_m\)’s, the estimates in \(W^{2,q}(\Sigma )\) are actually in \(C^{2,\alpha }_{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{m=1}^Mp_m\right)\) and the estimates in \(W^{2,q}_{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j'=1}^N\mathcal S_{j'}\right)\) are actually in \(C^{2,\alpha }_{\mathrm {loc}}\left(\Sigma {{\setminus }}\left(\mathop \bigcup \nolimits _{j'=1}^N\mathcal S_{j'}\cup \mathop \bigcup \nolimits _{m=1}^Mp_m\right)\right)\). Anyway, estimates in \(W^{2,q}\) will suffice in most of the paper.

To prove Theorem 3.1 we need two preliminary lemmas.

The first is a Harnack-type alternative for sequences of solutions of PDEs. It is inspired by [4, 17].

Lemma 3.2

Let \(\Omega \subset \Sigma \) be a connected open subset, \(\{f^n\}_{n\in \mathbb {N}}\) a bounded sequence in \(L^q_{\mathrm {loc}}(\Omega )\cap L^1(\Omega )\) for some \(q>1\) and \(\{w^n\}_{n\in \mathbb {N}}\) bounded from above and solving \(-\Delta w^n=f^n\) in \(\Omega \).

Then, up to subsequences, one of the following alternatives holds:

  • \(w^n\) is uniformly bounded in \(L^\infty _{\mathrm {loc}}(\Omega )\).

  • \(w^n\underset{n\rightarrow +\infty }{\longrightarrow }-\infty \) in \(L^\infty _{\mathrm {loc}}(\Omega )\).

Proof

Take a compact set \(\mathcal K\Subset \Omega \) and cover it with balls of radius \(\frac{r}{2}\), with r smaller than the injectivity radius of \(\Sigma \). By compactness, we can write \(\mathcal K\subset \mathop \bigcup \nolimits _{h=1}^HB_\frac{r}{2}(x_h)\). If the second alternative does not occur, then up to relabeling we get \(\sup _{B_r(x_1)}w^n\ge -C\).

Then, we consider the solution \(z^n\) of

$$\begin{aligned} \left\{ \begin{array}{ll}-\Delta z^n=f^n&{}\text{ in } B_r(x_1)\\ z^n=0&{}\text{ on } \partial B_r(x_1)\end{array}\right. , \end{aligned}$$

which is bounded in \(L^\infty (B_r(x_1))\) by elliptic estimates. This means that, for a large constant C, the function \(C-w^n+z^n\) is positive, harmonic and bounded from below on \(B_r(x_1)\), and moreover its infimum is bounded from above; therefore, applying the Harnack inequality (which is allowed since r is small enough) we get that \(C-w^n+z^n\) is uniformly bounded in \(L^\infty \left(B_\frac{r}{2}(x_1)\right)\), hence \(w^n\) is.

At this point, by connectedness, we can relabel the index h in such a way that \(B_\frac{r}{2}(x_h)\cap B_\frac{r}{2}(x_{h+1})\ne \emptyset \) for any \(h\in \{1,\dots ,H-1\}\) and we repeat the argument for \(B_\frac{r}{2}(x_2)\): since it has nonempty intersection with \(B_\frac{r}{2}(x_1)\), we have \(\sup _{B_r(x_2)}w^n\ge -C\), hence we get boundedness in \(L^\infty \left(B_\frac{r}{2}(x_2)\right)\). In the same way, we obtain the same result in all the balls \(B_\frac{r}{2}(x_h)\), whose union contains \(\mathcal K\), therefore \(w^n\) must be uniformly bounded on \(\mathcal K\) and we get the conclusion. \(\square \)

The second Lemma basically says that if all the concentration values in a point are under a certain threshold, and in particular if all of them equal zero, then compactness occurs around that point.

On the other hand, if a point belongs to some set \(\mathcal S_i\), then at least a fixed amount of mass has to accumulate around it; hence, being the total mass uniformly bounded from above, this can occur only for a finite number of points, so we deduce the finiteness of the \(\mathcal S_i\)’s.

Precisely, we have the following, inspired again by [17], Lemma 4.4:

Lemma 3.3

Let \(\left\{ v^n\right\} _{n\in \mathbb {N}}\) and \(\mathcal S_i\) be as in (13) and \(\sigma _i\) as in (10), and suppose \(\sigma _i(x)<\sigma _i^0\) for any \(i\in \{1,\dots ,N\}\), where

$$\begin{aligned} \sigma _i^0:=\frac{4\pi \min \left\{ 1,1+\min _{j\in \{1,\dots ,N\},m\in \{1,\dots ,M\}}\alpha _{jm}\right\} }{\sum _{j=1}^Na_{ij}^+}. \end{aligned}$$

Then, \(x\not \in \mathcal S_i\) for any \(i\in \{1,\dots ,N\}\).

Proof

First of all we notice that \(\sigma _i^0\) is well-defined for any i because \(a_{ii}>0\), hence \(\sum _{j=1}^Na_{ij}^+>0\).

Under the hypotheses of the Lemma, for large n and small r we have

$$\begin{aligned} \int _{B_r(x)}\widetilde{h}_i^ne^{v_i^n}{\mathrm {d}}V_g<\sigma _i^0. \end{aligned}$$
(14)

Let us consider \(w_i^n\) and \(z_i^n\) defined by

$$\begin{aligned} \left\{ \begin{array}{ll}-\Delta w_i^n=-\mathop \sum \limits _{j=1}^Na_{ij}\rho _j^n&{}\text{ in } B_r(x)\\ w_i^n=0&{}\text{ on } \partial B_r(x)\end{array}\right. ,\quad \quad \left\{ \begin{array}{ll}-\Delta z_i^n=\mathop \sum \limits _{j=1}^Na_{ij}^+\widetilde{h}_j^ne^{v_j^n}&{}\text{ in } B_r(x)\\ z_i^n=0&{}\text{ on } \partial B_r(x)\end{array}\right. . \end{aligned}$$
(15)

Is it evident that the \(w_i^n\)’s are uniformly bounded in \(L^\infty (B_r(x))\).

As for the \(z_i^n\)’s, we can suppose to be working on a Euclidean disc, up to applying a perturbation to \(\widetilde{h}_i^n\) which is smaller as r is smaller, hence for r small enough we still have the strict estimate (14).

Therefore, we get

$$\begin{aligned} \left\Vert-\Delta z_i^n\right\Vert_{L^1(B_r(x))}=\sum _{j=1}^Na_{ij}^+ \int _{B_r(x)}\widetilde{h}_j^ne^{v_j^n}{\mathrm {d}}V_g<\sum _{j=1}^Na_{ij}^+ \sigma _j^0\le 4\pi \min \{1,1+\alpha _i(x)\} \end{aligned}$$

and we can apply Lemma 2.1 to obtain \(\int _{B_r(x)}e^{q|z_i^n|}{\mathrm {d}}V_g\le C\) for some \(q>\frac{1}{\min \{1,1+\alpha _i(x)\}}\).

If \(\alpha _i(x)\ge 0\), then taking \(q\in \left(1,\frac{4\pi }{\left\Vert-\Delta z_i^n\right\Vert_{L^1(B_r(x))}}\right)\) we have

$$\begin{aligned} \int _{B_r(x)}\left(\widetilde{h}_i^ne^{z_i^n}\right)^q{\mathrm {d}}V_g\le C_r\int _{B_r(x)}e^{q|z_i^n|}{\mathrm {d}}V_g\le C. \end{aligned}$$

On the other hand, if \(\alpha _i(x)<0\), we choose

$$\begin{aligned} q\in \left(1,\frac{4\pi }{\left\Vert-\Delta z_i^n\right\Vert_{L^1(B_r(x))}-4\pi \alpha _i(x)}\right) \quad \quad \quad q'\in \left(\frac{4\pi }{4\pi -q\left\Vert-\Delta z_i^n\right\Vert_{L^1(B_r(x))}}, \frac{1}{-\alpha _i(x)q}\right) \end{aligned}$$

and, applying Hölder’s inequality,

$$\begin{aligned} \int _{B_r(x)}\left(\widetilde{h}_i^ne^{z_i^n}\right)^q{\mathrm {d}}V_g\le & {} C_r\int _{B_r(x)}d(\cdot ,x)^{2q\alpha _i(x)}e^{qz_i^n}{\mathrm {d}}V_g\\\le & {} C\left(\int _{B_r(x)}d(\cdot ,x)^{2qq'\alpha _i(x)}{\mathrm {d}}V_g\right)^\frac{1}{q'}\left(\int _{B_r(x)}e^{q\frac{q'}{q'-1}|z_i^n|}{\mathrm {d}}V_g\right)^{1-\frac{1}{q'}}\\\le & {} C, \end{aligned}$$

because \(qq'\alpha _i(x)>-1\) and \(q\frac{q'}{q'-1}\alpha _i(x)<\frac{4\pi }{\left\Vert-\Delta z_i^n\right\Vert_{L^1(B_r(x))}}\). Hence \(\widetilde{h}_i^ne^{z_i^n}\) is uniformly bounded in \(L^q(B_r(x))\) for some \(q>1\).

Now, let us consider \(v_i^n-z_i^n-w_i^n\): it is a subharmonic sequence by construction, so for any \(y\in B_\frac{r}{2}(x)\) we get

Moreover, since the maximum principle yields \(z_i^n \!\ge \! 0\), taking \(\theta \!=\!\small {\left\{ \begin{array}{ll}1&{}\text{ if } \alpha _i(x)\le 0\\ \in \left(0,\frac{1}{1+\alpha _i(x)}\right)&{}\text{ if } \alpha _i(x)>0\end{array}\right. }\), we get

$$\begin{aligned} \int _{B_r(x)}\left(v_i^n-z_i^n\right)^+{\mathrm {d}}V_g\le & {} \int _{B_r(x)}(v_i^n)^+{\mathrm {d}}V_g\\\le & {} \frac{1}{e\theta }\int _{B_r(x)}e^{\theta v_i^n}{\mathrm {d}}V_g\\\le & {} C\left\Vert\left(\widetilde{h}_i^n\right)^{-\theta }\right\Vert_{L^\frac{1}{1-\theta }(B_r(x))}\left(\int _{B_r(x)}\widetilde{h}_i^n e^{v_i^n}{\mathrm {d}}V_g\right)^\theta \\\le & {} C. \end{aligned}$$

Therefore, we showed that \(v_i^n-z_i^n-w_i^n\) is bounded from above in \(B_{\frac{r}{2}}(x)\), that is \(e^{v_i^n-z_i^n-w_i^n}\) is uniformly bounded in \(L^\infty \left(B_\frac{r}{2}(x)\right)\). Since the same holds for \(e^{w_i^n}\) and \(\widetilde{h}_i^ne^{z_i^n}\) is uniformly bounded in \(L^q\left(B_\frac{r}{2}(x)\right)\) for some \(q>1\), we deduce that also

$$\begin{aligned} \widetilde{h}_i^ne^{v_i^n}=\widetilde{h}_i^ne^{z_i^n}\,e^{v_i^n-z_i^n-w_i^n}\,e^{w_i^n} \end{aligned}$$

is bounded in the same \(L^q\left(B_\frac{r}{2}(x)\right)\).

Thus, we have an estimate on \(\left\Vert-\Delta z_i^n\right\Vert_{L^q\left(B_\frac{r}{2}(x)\right)}\) for any \(i\in \{1,\dots ,N\}\), hence by standard elliptic estimates we deduce that \(z_i^n\) is uniformly bounded in \(L^\infty \left(B_\frac{r}{2}(x)\right)\). Therefore, we also deduce that

$$\begin{aligned} v_i^n=\left(v_i^n-z_i^n-w_i^n\right)+z_i^n+w_i^n \end{aligned}$$

is bounded from above on \(B_\frac{r}{2}(x)\), which is equivalent to saying \(x\not \in \mathop \bigcup \nolimits _{i=1}^N\mathcal S_i\). \(\square \)

From this proof, we notice that, under the assumptions of Theorem 1.3, the same result holds for any single index \(i\in \{1,\dots ,N\}\). In other words, the upper bound on one \(\sigma _i\) implies that \(x\not \in \mathcal S_i\).

Corollary 3.4

Suppose \(a_{ij}\le 0\) for any \(i\ne j\).

Then, for any given \(i\in \{1,\dots ,N\}\) the following conditions are equivalent:

  • \(x\in \mathcal S_i\).

  • \(\sigma _i(x)\ne 0\).

  • \(\sigma _i(x)\ge \sigma _i'=\frac{4\pi \min \left\{ 1,1+\min _m\alpha _{im}\right\} }{a_{ii}}\).

Proof

The third statement trivially implies the second and the second implies the first, since if \(v_i^n\) is bounded from above in \(B_r(x)\) then \(\widetilde{h}_i^ne^{v_i^n}\) is bounded in \(L^q(B_r(x))\). Finally, if \(\sigma _i(x)<\sigma _i'\) then the sequence \(\widetilde{h}_i^ne^{z_i^n}\) defined by (15) is bounded in \(L^q\) for \(q>1\),so one can argue as in Lemma 3.3 to get boundedness from above of \(v_i^n\) around x, that is \(x\not \in \mathcal S_i\). \(\square \)

We can now prove the main theorem of this Section.

Proof of Theorem 3.1

If \(\mathcal S_i=\emptyset \) for any i, then \(e^{v_i^n}\) is bounded in \(L^\infty (\Sigma )\), so \(-\Delta v_i^n\) is bounded in \(L^q(\Sigma )\) for any

$$\begin{aligned} q\in \left[ 1,\frac{1}{-\min _{j\in \{1,\dots ,N\},m\in \{1,\dots ,M\}}\alpha _{jm}}\right) . \end{aligned}$$

Therefore, we can apply Lemma 3.2 to \(v_i^n\) on \(\Sigma \), where we must have the first alternative for every i, since otherwise the dominated convergence would give \(\int _\Sigma \widetilde{h}_i^ne^{v_i^n}\mathrm {d}V_g\underset{n\rightarrow +\infty }{\longrightarrow }0\) which is absurd; standard elliptic estimates allow to conclude compactness in \(W^{2,q}(\Sigma )\).

Suppose now \(\mathcal S_i\ne \emptyset \) for some i; from Lemma 3.3 we deduce

$$\begin{aligned} |\mathcal S_i|\sigma _i^0\le \sum _{x\in \mathcal S_i}\max _j\sigma _j(x)\le \sum _{j=1}^N\sum _{x\in \mathcal S_i}\sigma _j(x)\le \sum _{j=1}^N\rho _j, \end{aligned}$$

hence \(\mathcal S_i\) is finite.

For any \(j\in \{1,\dots ,N\}\), we can apply Lemma 3.2 on \(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j'=1}^N\mathcal S_{j'}\) with \(f^n=\sum _{j'=1}^Na_{jj'}\left(\widetilde{h}_{j'}^ne^{v_{j'}^n}-\rho _{j'}^n\right)\), since the last function is bounded in \(L^q_{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j'=1}^N\mathcal S_{j'}\right)\).

Therefore, either \(v_j^n\) goes to \(-\infty \) or it is bounded in \(L^\infty _{\mathrm {loc}}\), and in the last case we get compactness in \(W^{2,q}_{\mathrm {loc}}\) by applying again standard elliptic regularity. \(\square \)

4 Proof of Theorem 1.1

Here we will prove the theorem which gives sufficient and necessary conditions for the functional \(J_\rho \) to be bounded from below.

In other words, setting

$$\begin{aligned} E:=\left\{ \rho \in \mathbb {R}_+^N:\,J_\rho \text{ is } \text{ bounded } \text{ from } \text{ below } \text{ on } H^1(\Sigma )^N\right\} , \end{aligned}$$
(16)

we will prove that \(\left\{ \Lambda >0\right\} \subset E\subset \left\{ \Lambda \ge 0\right\} \).

As a first thing, we notice that the set E is not empty and it verifies a simple monotonicity condition.

Lemma 4.1

The set E defined by (16) is nonempty.

Moreover, for any \(\rho \in E\) then \(\rho '\in E\) provided \(\rho '_i\le \rho _i\) for any \(i\in \{1,\dots ,N\}\).

Proof

Let \(\theta >0\) be the biggest eigenvalue of the matrix \((a_{ij})\). Then,

$$\begin{aligned} J_\rho (u)\ge \sum _{i=1}^N\left(\frac{1}{2\theta }\int _\Sigma |\nabla u_i|^2{\mathrm {d}}V_g-\rho _i\left(\log \int _\Sigma \widetilde{h}_ie^{u_i}{\mathrm {d}}V_g-\overline{u_i}\right)\right). \end{aligned}$$

Therefore, from scalar Moser–Trudinger inequality (3), we deduce that \(J_\rho \) is bounded from below if \(\rho _i\le \frac{8\pi (1+\widetilde{\alpha }_i)}{\theta }\), hence \(E\ne \emptyset \).

Suppose now \(\rho \in E\) and \(\rho '_i\le \rho _i\) for any i. Then, through Jensen’s inequality, we get

$$\begin{aligned} J_{\rho '}(u)= & {} J_\rho (u)+\sum _{i=1}^N(\rho _i-\rho '_i)\log \int _\Sigma e^{u_i-\overline{u_i}+\log \widetilde{h}_i}{\mathrm {d}}V_g\\\ge & {} -C+\sum _{i=1}^N(\rho _i-\rho '_i)\int _\Sigma \log \widetilde{h}_i{\mathrm {d}}V_g\\\ge & {} -C \end{aligned}$$

for any \(u\in H^1(\Sigma )^N\), hence the claim. \(\square \)

It is interesting to observe that a similar monotonicity condition is also satisfied by the set \(\{\Lambda >0\}\) (although one can easily see that it is not true if we replace \(\Lambda \) with \(\Lambda _{\mathcal I,x}\)).

Lemma 4.2

Let \(\rho ,\rho '\in \mathbb {R}_+^N\) be such that \(\Lambda (\rho )>0\) and \(\rho '_i\le \rho _i\) for any \(i\in \{1,\dots ,N\}\).

Then, \(\Lambda (\rho ')>0\).

Proof

Suppose by contradiction \(\Lambda (\rho ')\le 0\), that is \(\Lambda _{\mathcal I,x}(\rho ')\le 0\) for some \({\mathcal {I}},x\).

This cannot occur for \({\mathcal {I}}=\{i\}\) because it would mean \(\rho '_i\ge \frac{8\pi (1+\alpha _i(x))}{a_{ii}}\), so the same inequality would for \(\rho _i\), hence \(\Lambda (\rho )\le \Lambda _{{\mathcal {I}},x}(\rho )\le 0\).

Therefore, there must be some \({\mathcal {I}},x\) such that \(\Lambda _{\mathcal I,x}(\rho ')\le 0\) and \(\Lambda _{{\mathcal {I}}{{\setminus }}\{i\},x}(\rho ')>0\) for any \(i\in {\mathcal {I}}\); this implies

$$\begin{aligned} 0< & {} \Lambda _{{\mathcal {I}}{{\setminus }}\{i\},x}(\rho ')-\Lambda _{{\mathcal {I}},x}(\rho ')\nonumber \\= & {} 2\sum _{j\in {\mathcal {I}}}a_{ij}\rho '_i\rho '_j-a_{ii}{\rho '_i}^2-8\pi (1+\alpha _i(x))\rho '_i\nonumber \\< & {} \rho '_i\left(2\sum _{j\in {\mathcal {I}}}a_{ij}\rho '_j-8\pi (1+\alpha _i(x))\right). \end{aligned}$$
(17)

It will be not restrictive to suppose, from now on, \(\rho _1'\le \rho _1\) and \(\rho _i'=\rho _i\) for any \(i\ge 2\), since the general case can be treated by exchanging the indices and iterating.

Assuming this, we must have \(1\in {\mathcal {I}}\), therefore we obtain:

$$\begin{aligned} 0< & {} \Lambda _{{\mathcal {I}},x}(\rho )-\Lambda _{{\mathcal {I}},x}(\rho ')\\= & {} 8\pi (1+\alpha _1(x))(\rho _1-\rho _1')-a_{11}\left({\rho _1'}^2-\rho _1^2\right)-2\sum _{j\in {\mathcal {I}}{{\setminus }}\{1\}}a_{1j}(\rho _1'-\rho _1)\rho _j\\= & {} (\rho _1-\rho '_1)\left(8\pi (1+\alpha _1(x))-a_{11}(\rho _1'+\rho _1)-2\sum _{j\in {\mathcal {I}}{{\setminus }}\{1\}}a_{1j}\rho _j\right)\\< & {} (\rho _1-\rho '_1)\left(8\pi (1+\alpha _1(x))-2\sum _{j\in {\mathcal {I}}}a_{1j}\rho '_j\right), \end{aligned}$$

which is negative by (17). We found a contradiction. \(\square \)

We will now show that if the parameter \(\rho \) lies in the interior of E then not only the functional is bounded from below but it is coercive in the space of zero-average functions. In particular, this fact allows to deduce the “if” part in Corollary 1.2 from Theorem 1.1.

On the other hand, if \(\rho \) belongs to the boundary of E, then the scenario is quite different.

Lemma 4.3

Suppose \(\rho \in \mathring{E}\). Then, there exists a constant \(C=C_\rho \) such that

$$\begin{aligned} J_\rho (u)\ge \frac{1}{C}\sum _{i=1}^N\int _\Sigma |\nabla u_i|^2{\mathrm {d}}V_g-C. \end{aligned}$$

Moreover, \(J_\rho \) admits a minimizer which solves (6).

Proof

Choosing \(\delta \in \left(0,\frac{d(\rho ,\partial E)}{\sqrt{N}|\rho |}\right)\) one has \((1+\delta )\rho \in E\), so

$$\begin{aligned} J_\rho (u)= & {} \frac{\delta }{2(1+\delta )}\sum _{i,j=1}^Na^{ij}\int _\Sigma \nabla u_i\cdot \nabla u_j{\mathrm {d}}V_g+\frac{1}{1+\delta }J_{(1+\delta )\rho }(u)\\\ge & {} \frac{\delta }{2\theta (1+\delta )}\sum _{i=1}^N\int _\Sigma |\nabla u_i|^2{\mathrm {d}}V_g-C, \end{aligned}$$

hence we get the former claim.

To get the latter, we notice that, due to invariance by translation, any minimizer can be supposed to be in \(\overline{H}^1(\Sigma )^N\); therefore, we can restrict \(J_\rho \) to this subspace. Here, the above inequality implies coercivity, and it is immediate to see that \(J_\rho \) is also lower semi-continuous, hence the existence of minimizers follows from direct methods of calculus of variations. \(\square \)

Lemma 4.4

Suppose \(\rho \in \partial E\). Then, there exists a sequence \(\left\{ u^n\right\} _{n\in \mathbb {N}}\subset H^1(\Sigma )^N\) such that

$$\begin{aligned} \sum _{i=1}^N\int _\Sigma \left|\nabla u_i^n\right|^2{\mathrm {d}}V_g\underset{n\rightarrow +\infty }{\longrightarrow }+\infty \quad \quad \quad \lim _{n\rightarrow +\infty }\frac{J_\rho \left(u^n\right)}{\sum _{i=1}^N\int _\Sigma \left|\nabla u_i^n\right|^2{\mathrm {d}}V_g}\le 0 \end{aligned}$$

Proof

We first notice that \((1-\delta )\rho \in E\) for any \(\delta \in (0,1)\). In fact, otherwise, from Lemma 4.1 we would get \(\rho '\not \in E\) as soon as \(\rho '_i\ge (1-\delta )\rho _i\) for some i, hence \(\rho \not \in \partial E\).

Now, suppose by contradiction that for any sequence \(u^n\) one gets

$$\begin{aligned} \sum _{i=1}^N\int _\Sigma \left|\nabla u_i^n\right|^2{\mathrm {d}}V_g\underset{n\rightarrow +\infty }{\longrightarrow }+\infty \quad \quad \quad \Rightarrow \quad \quad \quad \frac{J_\rho \left(u^n\right)}{\sum _{i=1}^N\int _\Sigma \left|\nabla u_i^n\right|^2{\mathrm {d}}V_g}\ge \varepsilon >0. \end{aligned}$$

Therefore, we would have

$$\begin{aligned} J_\rho (u)\ge \frac{\varepsilon }{2}\sum _{i=1}^N\int _\Sigma |\nabla u_i|^2{\mathrm {d}}V_g-C; \end{aligned}$$

hence, indicating as \(\theta '\) the smallest eigenvalue of the matrix A, for small \(\delta \) we would get

$$\begin{aligned} J_\rho (u)= & {} (1+\delta )J_{(1+\delta )\rho }(u)-\frac{\delta }{2}\sum _{i,j=1}^Na^{ij}\int _\Sigma \nabla u_i\cdot \nabla u_j{\mathrm {d}}V_g\\\ge & {} \left((1+\delta )\frac{\varepsilon }{2}-\frac{\delta }{2\theta '}\right)\sum _{i=1}^N\int _\Sigma |\nabla u_i|^2-C\\\ge & {} -C. \end{aligned}$$

So we obtain \((1+\delta )\rho \in E\); being also \((1-\delta )\rho \in E\) (by Lemma 4.1), we get a contradiction with \(\rho \in \partial E\). \(\square \)

To see what happens when \(\rho \in \partial E\), we build an auxiliary functional using Lemma 2.3.

Lemma 4.5

Fix \(\rho '\in \partial E\) and define:

$$\begin{aligned} a_{\rho '}^n:= & {} \frac{1}{2}\sum _{i,j=1}^Na^{ij}\int _\Sigma \nabla u_i^n\cdot \nabla u_j^n {\mathrm {d}}V_g\quad \quad \quad b_{\rho '}^n:=J_{\rho '}\left(u^n\right)\\ J'_{\rho ',\rho }(u)= & {} J_\rho (u)-F_{\rho '}\left(\frac{1}{2}\sum _{i,j=1}^Na^{ij} \int _\Sigma \nabla u_i\cdot \nabla u_j{\mathrm {d}}V_g\right), \end{aligned}$$

where \(u^n\) is given by Lemma 4.4 and \(F_{\rho '}\) by Lemma 2.3.

If \(\rho \in \mathring{E}\), then \(J'_{\rho ',\rho }\) is bounded from below on \(H^1(\Sigma )^N\) and its infimum is achieved by a solution of

$$\begin{aligned} - \Delta \left(u_i-\sum _{i,j=1}^Na^{ij}fu_j\right)=\sum _{j=1}^Na_{ij}\rho _j \left(\frac{\widetilde{h}_je^{u_j}}{\int _\Sigma \widetilde{h}_je^{u_j}{\mathrm {d}}V_g}-1\right),\quad \quad \quad i=1,\dots ,N, \end{aligned}$$

with \(f=\left(F_{\rho '}\right)'\left(\frac{1}{2}\mathop \sum \nolimits _{i,j=1}^Na^{ij}\int _\Sigma \nabla u_i\cdot \nabla u_j{\mathrm {d}}V_g\right)\).

On the other hand, \(J'_{\rho ',\rho '}\) is unbounded from below.

Proof

For \(\rho \in \mathring{E}\), we can argue as in Lemma 4.3, since the continuity follows from the regularity of F and the coercivity from the behavior of \(F'\) at the infinity.

For \(\rho =\rho '\), if we take \(u^n\) as in Lemma 4.4 we get

$$\begin{aligned} J'_{\rho ',\rho '}\left(u^n\right)=b_{\rho '}^n-F_{\rho '}\left(a_{\rho '}^n\right)\underset{n\rightarrow +\infty }{\longrightarrow }-\infty . \end{aligned}$$

\(\square \)

Now we can prove the first half of Theorem 1.1, that is \(J_\rho \) is bounded from below if \(\Lambda (\rho )>0\).

Proof of \(\left\{ \Lambda >0\right\} \subset E\) Suppose by contradiction there is some \(\rho '\in \partial E\) with \(\Lambda (\rho )>0\) and take a sequence \(\rho ^n\in E\) with \(\rho ^n\underset{n\rightarrow +\infty }{\longrightarrow }\rho '\).

Then, by Lemma 4.5, the auxiliary functional \(J_{\rho ',\rho ^n}\) admits a minimizer \(u^n\), so the functions \(v_i^n\) defined as in (11) solve

$$\begin{aligned} \left\{ \begin{array}{l}-\Delta v_i^n=\mathop \sum \limits _{j,j'=1}^Na_{ij}b^{jj',n}\left(\widetilde{h}_je^{v_j^n}-\rho _j^n\right)\\ \int _\Sigma \widetilde{h}_i^ne^{v_i^n}{\mathrm {d}}V_g=\rho _i^n\end{array}\right. \quad \quad \quad i=1,\dots ,N \end{aligned}$$

where \(b^{ij,n}\) is the inverse matrix of \(b_{ij}^n:=\delta _{ij}-a^{ij}f^n\), hence \(b^{ij,n}\underset{n\rightarrow +\infty }{\longrightarrow }\delta _{ij}\).

We can then apply Theorem 3.1. The first alternative is excluded, since otherwise we would get, for any \(u\in H^1(\Sigma )^N\),

$$\begin{aligned} J'_{\rho ',\rho '}(u)=\lim _{n\rightarrow +\infty }J'_{\rho ',\rho ^n}(u)\ge \lim _{n\rightarrow +\infty }J'_{\rho ',\rho ^n}\left(v^n\right)=J'_{\rho ',\rho '}(v)>-\infty , \end{aligned}$$

thus contradicting Lemma 4.5.

Therefore, blow up must occur; this means, by Lemma 3.3, that \(\sigma _i(p)\ne 0\) for some \(i\in \{1,\dots ,N\}\) and some \(p\in \Sigma \).

By Proposition 2.2 follows \(\Lambda (\sigma )\le 0\). On the other hand, since \(\sigma _i\le \rho '_i\) for any i, Lemma 4.2 yields \(\Lambda (\rho ')\le 0\), which contradicts our assumptions. \(\square \)

To prove the unboundedness from below of \(J_\rho \) in the case \(\Lambda (\rho )<0\) we will use suitable test functions, whose properties are described by the following:

Lemma 4.6

Define, for \(x\in \Sigma \) and \(\lambda >0,\,\varphi =\varphi ^{\lambda ,x}\) as

$$\begin{aligned} \varphi _i:=-2(1+\alpha _i(x))\log \max \{1,\lambda d(\cdot ,x)\}. \end{aligned}$$

Then, as \(\lambda \rightarrow +\infty \), one has

$$\begin{aligned}&\int _\Sigma \nabla \varphi _i\cdot \nabla \varphi _j{\mathrm {d}}V_g=8\pi (1+\alpha _i(x))(1+\alpha _j(x))\log \lambda +O(1)\\&\quad \overline{\varphi _i}=-2(1+\alpha _i(x))\log \lambda +O(1)\\&\quad \int _\Sigma \widetilde{h}_ie^{\sum _{j=1}^N\theta _j\varphi _j}{\mathrm {d}}V_g\ge C\lambda ^{-2(1+\alpha _i(x))}\quad \quad \quad \text{ if }\quad \sum _{i=1}^N\theta _j(1+\alpha _j(x))>1+\alpha _i(x). \end{aligned}$$

Proof

It holds

$$\begin{aligned} \nabla \varphi _i=\left\{ \begin{array}{l@{\quad }l}0&{}\text{ if } d(\cdot ,x)<\frac{1}{\lambda }\\ -2(1+\alpha _i(x))\frac{\nabla d(\cdot ,x)}{d(\cdot ,x)}&{}\text{ if } d(\cdot ,x)>\frac{1}{\lambda }\end{array}\right. . \end{aligned}$$

Therefore, being \(|\nabla d(\cdot ,x)|=1\) almost everywhere on \(\Sigma \):

$$\begin{aligned}&\int _\Sigma \nabla \varphi _i\cdot \nabla \varphi _j{\mathrm {d}}V_g\\&\quad =4(1+\alpha _i(x))(1+\alpha _j(x))\int _{\Sigma {{\setminus }} B_\frac{1}{\lambda }(x)}\frac{{\mathrm {d}}V_g}{d(\cdot ,x)^2}\\&\quad =8\pi (1+\alpha _i(x))(1+\alpha _j(x))\log \lambda +O(1). \end{aligned}$$

For the average of \(\varphi _i\), we get

$$\begin{aligned} \int _\Sigma \varphi _i{\mathrm {d}}V_g= & {} -2(1+\alpha _i(x))\int _{\Sigma {{\setminus }} B_\frac{1}{\lambda }(x)}(\log \lambda +\log d(\cdot ,x)){\mathrm {d}}V_g+O(1)\\= & {} -2(1+\alpha _i(x))\log \lambda +O(1). \end{aligned}$$

For the last estimate, choose \(r>0\) such that \(\overline{B_\delta (x)}\) does not contain any of the points \(p_m\) for \(m=1,\dots ,M\), except possibly x.

Then, outside such a ball, \(e^{\sum _{j=1}^N\theta _j\varphi _j}\le C\lambda ^{-2\sum _{j=1}^N\theta _j(1+\alpha _j(x))}\).

Therefore, under the assumptions of the Lemma,

$$\begin{aligned} \int _{\Sigma {{\setminus }} B_\delta (x)}\widetilde{h}_ie^{\sum _{i=1}^N\theta _j\varphi _j}{\mathrm {d}}V_g=o\left(\lambda ^{-2(1+\alpha _i(x))}\right), \end{aligned}$$

hence

$$\begin{aligned} \int _\Sigma \widetilde{h}_ie^{\sum _{i=1}^N\theta _j\varphi _j}\mathrm dV_g\ge & {} \int _{B_\delta (x)}\widetilde{h}_ie^{\sum _{i=1}^N\theta _j\varphi _j}\mathrm dV_g\\\ge & {} C\left(\int _{B_\frac{1}{\lambda }(x)}d(\cdot ,x)^{2\alpha _i(x)}\mathrm dV_g+\frac{1}{\lambda ^{2\sum _{j=1}^N\theta _j(1+\alpha _j(x))}}\right.\\&\left.\int _{A_{\frac{1}{\lambda },\delta }(x)}d(\cdot ,x)^{2\alpha _i(x)-2\sum _{i=1}^N\theta _j(1+\alpha _j(x))}\mathrm dV_g\right)\\\ge & {} C\lambda ^{-2(1+\alpha _i(x))}, \end{aligned}$$

which concludes the proof. \(\square \)

Proof of \(E\subset \left\{ \Lambda \ge 0\right\} \) Take \(\rho ,{\mathcal {I}},x\) such that \(\Lambda _{{\mathcal {I}},x}(\rho )<0\) and \(\Lambda _{{\mathcal {I}}{{\setminus }}\{i\},x}(\rho )\ge 0\) for any \(i\in {\mathcal {I}}\), and consider the family of functions \(\left\{ u^\lambda \right\} _{\lambda >0}\) defined by

$$\begin{aligned} u_i^\lambda :=\sum _{j\in {\mathcal {I}}}\frac{a_{ij}\rho _j}{4\pi (1+\alpha _i(x))}\varphi _j^{\lambda ,x}. \end{aligned}$$

By Jensen’s inequality we get

$$\begin{aligned} J_\rho \left(u^\lambda \right)\le & {} \frac{1}{2}\sum _{i,j=1}^Na^{ij}\int _\Sigma \nabla u_i^\lambda \cdot \nabla u_j^\lambda {\mathrm {d}}V_g+\sum _{i\in {\mathcal {I}}}\rho _i\left(\overline{u_i^\lambda }-\log \int _\Sigma \widetilde{h}_ie^{u_i^\lambda }{\mathrm {d}}V_g\right)+C\\= & {} \frac{1}{2}\sum _{i,j\in {\mathcal {I}}}\frac{a_{ij}\rho _i\rho _j}{16\pi ^2(1+\alpha _i(x))(1+\alpha _j(x))}\int _\Sigma \nabla \varphi _i\cdot \nabla \varphi _j{\mathrm {d}}V_g\\&+\sum _{i,j\in {\mathcal {I}}}\frac{a_{ij}\rho _i\rho _j}{4\pi (1+\alpha _j(x))}\overline{\varphi _j}-\sum _{i\in {\mathcal {I}}}\rho _i\log \int _\Sigma \widetilde{h}_ie^{\sum _{j\in {\mathcal {I}}}\frac{a_{ij}\rho _j}{4\pi (1+\alpha _j(x))}\varphi _j}{\mathrm {d}}V_g+C. \end{aligned}$$

At this point, we would like to apply Lemma 4.6 to estimate \(J_\rho \left(u^\lambda \right)\). To be able to do this, we have to verify that

$$\begin{aligned} \frac{1}{4\pi }\sum _{j\in {\mathcal {I}}}a_{ij}\rho _j>1+\alpha _i(x)\quad \quad \quad \forall \,i\in {\mathcal {I}}. \end{aligned}$$

If \({\mathcal {I}}=\{i\}\), then \(\rho _i>\frac{8\pi (1+\alpha _i(x))}{a_{ii}}\), so it follows immediately. For the other cases, it follows from (17).

So we can apply Lemma 4.6 and we get from the previous estimates:

$$\begin{aligned}&J_\rho \left(u^\lambda \right) \le \left(\frac{1}{4\pi }\sum _{i,j\in {\mathcal {I}}}a_{ij}\rho _i\rho _j-\frac{1}{2\pi }\sum _{i,j\in {\mathcal {I}}}a_{ij}\rho _i\rho _j+2\sum _{i\in {\mathcal {I}}}\rho _i(1+\alpha _i(x))\right)\log \lambda +C\\&\quad =-\frac{\Lambda _{{\mathcal {I}},x}(\rho )}{4\pi }\log \lambda +C\underset{n\rightarrow +\infty }{\longrightarrow }-\infty . \end{aligned}$$

\(\square \)

Proof of Corollary 1.2

The coercivity in the case \(\Lambda <0\), hence the existence of minimizing solutions for (6) follows from Theorem 1.1 and Lemma 4.3.

If instead \(\Lambda (\rho )\ge 0\), then one can find out the lack of coercivity by arguing as before with the sequence \(u^\lambda \), which verifies

$$\begin{aligned} \sum _{i=1}^N\int _\Sigma \left|\nabla u_i^\lambda \right|^2{\mathrm {d}}V_g\underset{\lambda \rightarrow +\infty }{\longrightarrow }+\infty \quad \quad \quad J_\rho \left(u^\lambda \right)\le -\frac{\Lambda _{{\mathcal {I}},x}(\rho )}{4\pi }\log \lambda +C\le C. \end{aligned}$$

\(\square \)

5 Proof of Theorem 1.3

Here we will finally prove a sharp inequality in the case when the matrix \(a_{ij}\) has non-positive entries outside its main diagonal.

As already pointed out in the introduction, the function \(\Lambda (\rho )\) can be written in a much shorter form under these assumptions, so the condition \(\Lambda (\rho )\ge 0\) is equivalent to \(\rho _i\le \frac{8\pi (1+\widetilde{\alpha }_i)}{a_{ii}}\) for any \(i\in \{1,\dots ,N\}\).

Moreover, thanks to Lemma 4.1, in order to prove Theorem 1.3 for all such \(\rho \)’s it will suffice to consider

$$\begin{aligned} \rho ^0:=\left(\frac{8\pi (1+\widetilde{\alpha }_1)}{a_{11}},\dots ,\frac{8\pi (1+\widetilde{\alpha }_N)}{a_{NN}}\right). \end{aligned}$$
(18)

By what we proved in the previous Section, for any sequence \(\rho ^n\underset{n\rightarrow +\infty }{\nearrow }\rho ^0\) one has

$$\begin{aligned} \inf _{H^1(\Sigma )^N}J_{\rho ^n}=J_{\rho ^n}(u^n)\ge -C_{\rho ^n}, \end{aligned}$$

so Theorem 1.3 will follow by showing that, for a given sequence \(\left\{ \rho ^n\right\} _{n\in \mathbb {N}}\), the constant \(C_n=C_{\rho ^n}\) can be chosen independently of n.

As a first thing, we provide a Lemma which shows the possible blow-up scenarios for such a sequence \(u^n\).

Here, the assumption on \(a_{ij}\) is crucial since it reduces largely the possible cases.

Lemma 5.1

Let \(\rho ^0\) be as in (18), \(\left\{ \rho ^n\right\} _{n\in \mathbb {N}}\) such that \(\rho ^n\nearrow \rho ^0\), \(u^n\) a minimizer of \(J_{\rho ^n}\) and \(v^n\) as in (11). Then, up to subsequences, there exists a set \({\mathcal {I}}\subset \{1,\dots ,N\}\) such that:

  • If \(i\in {\mathcal {I}}\), then \(\mathcal S_i=\{x_i\}\) for some \(x_i\in \Sigma \) which satisfy \(\widetilde{\alpha }_i=\alpha _i(x_i)\) and \(\sigma _i(x_i)=\rho _i^0\), and \(v_i^n\underset{n\rightarrow +\infty }{\longrightarrow }-\infty \) in \(L^\infty _{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j\in {\mathcal {I}}}\{x_j\}\right)\).

  • If \(i\not \in {\mathcal {I}}\), then \(\mathcal S_i=\emptyset \) and \(v_i^n\underset{n\rightarrow +\infty }{\longrightarrow }v_i\) in \(W^{2,q}_{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j\in {\mathcal {I}}}\{x_j\}\right)\) for some \(q>1\) and some suitable \(v_i\).

Moreover, if \(a_{ij}<0\) then \(x_i\ne x_j\).

Proof

From Theorem 3.1 we get a \({\mathcal {I}}\subset \{1,\dots ,N\}\) such that \(\mathcal S_i\ne \emptyset \) for \(i\in {\mathcal {I}}\).

If \(\mathcal S_i\ne \emptyset \), then by Corollary 3.4 one gets

$$\begin{aligned} 0<\sigma _i(x)\le \rho _i^0\le \frac{8\pi (1+\alpha _i(x))}{a_{ii}} \end{aligned}$$

for all \(x\in \mathcal S_i\), hence

$$\begin{aligned} 0= & {} \Lambda _{\{1,\dots ,N\},x}(\sigma (x))\nonumber \\\ge & {} \sum _{j=1}^N\left(8\pi (1+\alpha _j(x))\sigma _j(x)-a_{jj}\sigma _j(x)^2\right)\nonumber \\\ge & {} 8\pi (1+\alpha _i(x))\sigma _i(x)-a_{ii}\sigma _i(x)^2\nonumber \\\ge & {} 0. \end{aligned}$$
(19)

Therefore, all these inequalities must actually be equalities.

From the last, we have \(\sigma _i(x)=\rho _i^0=\frac{8\pi (1+\alpha _i(x))}{a_{ii}}\), hence \(\alpha _i(x)=\widetilde{\alpha }_i\). On the other hand, since \(\mathop \sum \nolimits _{x\in \mathcal S_i}\sigma _i(x)\le \rho _i^0\), it must be \(\sigma _i(x)=0\) for all but one \(x_i\in \mathcal S_i\), so Corollary 3.4 yields \(\mathcal S_i=\{x_i\}\).

Let us now show that \(v_i^n\underset{n\rightarrow +\infty }{\longrightarrow }-\infty \) in \(L^\infty _{\mathrm {loc}}\).

Otherwise, Theorem 3.1 would imply \(v_i^n\underset{n\rightarrow +\infty }{\longrightarrow }v_i\) almost everywhere, therefore by Fatou’s Lemma we would get the following contradiction:

$$\begin{aligned} \sigma _i(x_i)<\int _\Sigma \widetilde{h}_ie^{v_i}{\mathrm {d}}V_g+\sigma _i(x_i)\le \int _\Sigma \widetilde{h}_i^ne^{v_i^n}{\mathrm {d}}V_g=\rho _i^n\le \rho _i=\sigma _i(x_i). \end{aligned}$$

Since also inequality (19) has to be an equality, we get \(a_{ij}\sigma _i(x_i)\sigma _j(x_i)\) for any \(i,j\in {\mathcal {I}}\), so whenever \(a_{ij}<0\) there must be \(\sigma _j(x_i)=0\), so \(x_i\ne x_j\).

Finally, if \(\mathcal S_i=\emptyset \), the convergence in \(W^{2,q}_{\mathrm {loc}}\) follows from what we just proved and Theorem 3.1. \(\square \)

We basically showed that if a component of the sequence \(v^n\) blows up, then all its mass concentrates at a single point which has the lowest singularity coefficient.

The next Lemma gives some more important information about the convergence or the blow-up of the components of \(v^n\).

Lemma 5.2

Let \(v_i^n,\,v_i,\,\rho ^0,\,{\mathcal {I}}\) and \(x_i\) as in Lemma 5.1.

Then,

  • If \(i\in {\mathcal {I}}\), then the sequence \(v_i^n-\overline{v_i^n}\) converges to some \(G_i\) in \(W^{2,q}_{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j\in {\mathcal {I}}}\{x_j\}\right)\) for some \(q>1\) and weakly in \(W^{1,q'}(\Sigma )\) for any \(q'\in (1,2)\), and \(G_i\) solves:

    $$\begin{aligned} \left\{ \begin{array}{l}-\Delta G_i=\mathop \sum \limits _{j\in {\mathcal {I}}}a_{ij}\rho _j^0\left(\delta _{x_j}-1\right)+\mathop \sum \limits _{j\not \in {\mathcal {I}}}a_{ij}\left(\widetilde{h}_je^{v_j}-\rho _j^0\right)\\ \overline{G_i}=0\end{array}\right. . \end{aligned}$$

  • If \(i\not \in {\mathcal {I}}\), then \(v_i^n\underset{n\rightarrow +\infty }{\longrightarrow }v_i\) in the same space, and \(v_i\) solves:

    $$\begin{aligned} \left\{ \begin{array}{l}-\Delta v_i=\mathop \sum \limits _{j\in {\mathcal {I}}}a_{ij}\rho _j^0\left(\delta _{x_j}-1\right)+\mathop \sum \limits _{j\not \in {\mathcal {I}}}a_{ij}\left(\widetilde{h}_je^{v_j}-\rho _j^0\right)\\ \int _\Sigma \widetilde{h}_ie^{v_i}{\mathrm {d}}V_g=\rho _i^0\end{array}\right. . \end{aligned}$$
    (20)

Proof

From Lemma 5.1 follows that, for \(i\in {\mathcal {I}}\), \(\widetilde{h}_i^ne^{v_i^n}\underset{n\rightarrow \infty }{\rightharpoonup }\rho _i^0\delta _{x_i}\) in the sense of measures; in fact, for any \(\phi \in C(\Sigma )\)

$$\begin{aligned} \left|\int _\Sigma \widetilde{h}_i^ne^{v_i^n}\phi {\mathrm {d}}V_g-\rho _i^0\phi (x_i)\right|\le & {} \int _\Sigma \widetilde{h}_i^ne^{v_i^n}|\phi -\phi (x_i)|{\mathrm {d}}V_g+\left|\rho _i^n-\rho _i^0\right||\phi (x_i)|\\\le & {} \varepsilon \int _{B_\delta (x_i)}\widetilde{h}_i^ne^{v_i^n}{\mathrm {d}}V_g+2\Vert \phi \Vert _{L^\infty (\Sigma )}\int _{\Sigma {{\setminus }} B_\delta (x_i)}\widetilde{h}_i^ne^{v_i^n}{\mathrm {d}}V_g\\&+\left|\rho _i^n-\rho _i^0\right|\Vert \phi \Vert _{L^\infty (\Sigma )}\\\le & {} \varepsilon \rho _i^n+2\Vert \phi \Vert _{L^\infty (\Sigma )}o(1)+o(1)\Vert \phi \Vert _{L^\infty (\Sigma )}, \end{aligned}$$

which is, choosing properly \(\varepsilon \), arbitrarily small. Therefore, \(v_i\) solves (20).

On the other hand, if \(q'\in (1,2)\), then \(\frac{q'}{q'-1}>2\), so any function \(\phi \in W^{1,\frac{q'}{q'-1}}(\Sigma )\) is actually continuous, hence

$$\begin{aligned}&\left|\int _\Sigma \nabla \left(v_i^n-\overline{v_i^n}-G_i\right)\cdot \nabla \phi {\mathrm {d}}V_g\right|\\&\quad =\left|\int _\Sigma \left(-\Delta v_i^n+\Delta G_i\right)\phi dV_h\right|\\&\quad \le \sum _{j\in {\mathcal {I}}}a_{ij}\left|\int _\Sigma \widetilde{h}_je^{v_j^n}\phi {\mathrm {d}}V_g-\rho _j^0\phi (p)\right|\\&\qquad +\sum _{j\not \in {\mathcal {I}}}a_{ij}\left|\int _\Sigma \widetilde{h}_j\left(e^{v_j^n}-e^{v_j}\right)\phi {\mathrm {d}}V_g\right|\underset{n\rightarrow +\infty }{\longrightarrow }0. \end{aligned}$$

Therefore, we get weak convergence in \(W^{1,q'}(\Sigma )\) for any \(q'\in (1,2)\); standard elliptic estimates yield convergence in \(W^{2,q}_{\mathrm {loc}}\left(\Sigma {{\setminus }}\mathop \bigcup \nolimits _{j\in {\mathcal {I}}}\{x_j\}\right)\).

In the same way we prove the same convergence of \(v_i^n\) to \(v_i\). \(\square \)

From these information about the blow-up profile of \(v^n\) we deduce an important fact which will be used to prove the main Theorem:

Corollary 5.3

Let \(v^n\) and \(x_i\) be as in Lemmas 5.1 and 5.2 and \(w^n\) be defined by \(w_i^n=\mathop \sum \nolimits _{j=1}^Na^{ij}v_j^n\) for \(i\in \{1,\dots ,N\}\).

Then, \(w_i^n-\overline{w_i^n}\) is uniformly bounded in \(W^{2,q}_{\mathrm {loc}}(\Sigma {{\setminus }}\{x_i\})\) for some \(q>1\) if \(i\in \mathcal I\), whereas if \(i\not \in {\mathcal {I}}\) it is bounded in \(W^{2,q}(\Sigma )\).

Proof

Since \(-\Delta w_i^n=\widetilde{h}_i^ne^{v_i^n}-\rho _i^n\), the claim follows from the boundedness of \(e^{v_i^n}\) in \(L^\infty _{\mathrm {loc}}(\Sigma {{\setminus }}\{x_i\})\) and from standard elliptic estimates. \(\square \)

The last Lemma we need is a localized scalar Moser–Trudinger inequality for the blowing-up sequence.

Lemma 5.4

Let \(w_i^n\) be as in Corollary 5.3 and \(x_i\) as in the previous Lemmas. Then, for any \(i\in {\mathcal {I}}\) and any small \(r>0\) one has

$$\begin{aligned} \frac{a_{ii}}{2}\int _{B_r(x_i)}\left|\nabla w_i^n\right|^2{\mathrm {d}}V_g-\rho _i^n\left(\log \int _{B_r(x_i)}\widetilde{h}_ie^{a_{ii}w_i^n}{\mathrm {d}}V_g-a_{ii}\overline{w_i^n}\right)\ge -C_r. \end{aligned}$$

Proof

Since \(\Sigma \) is locally conformally flat, we can choose r small enough so that we can apply Corollary 2.5 up to modifying \(\widetilde{h}_i^n\). We also take r so small that \(\overline{B_r(x_i)}\) contains neither any \(x_j\) for \(x_j\ne x_i\) nor any \(p_m\) for \(m=1,\dots ,M\) (except possibly \(x_i\)).

Let \(z^n\) be the solution of

$$\begin{aligned} \left\{ \begin{array}{ll}-\Delta z_i^n=\widetilde{h}_i^ne^{v_i^n}-\rho _i^n&{}\text{ in } B_r(x_i)\\ z_i^n=0&{}\text{ on } \partial B_r(x_i)\end{array}\right. . \end{aligned}$$

Then, \(w_i^n-\overline{w_i^n}-z_i^n\) is harmonic and it has the same value as \(w_i^n-\overline{w_i^n}\) on \(\partial B_r(x_i)\), so from standard estimates

$$\begin{aligned} \left\| w_i^n-\overline{w_i^n}-z_i^n\right\| _{C^1(B_r(x_i))}\le C \left\| w_i^n-\overline{w_i^n}\right\| _{C^1(\partial B_r(x_i))}\le C. \end{aligned}$$

From Lemma 5.2 we get

$$\begin{aligned} \left|\int _{B_r(x_i)}\left|\nabla w_i^n\right|^2{\mathrm {d}}V_g-\int _{B_r(x_i)}\left|\nabla z_i^n\right|^2{\mathrm {d}}V_g\right|= & {} \left|\int _{B_r(x_i)}\left|\nabla \left(w_i^n-z_i^n\right)\right|^2{\mathrm {d}}V_g\right.\\&+\left.2\int _{B_r(x_i)}\nabla w_i^n\cdot \nabla \left(w_i^n-z_i^n\right){\mathrm {d}}V_g\right|\\\le & {} \int _{B_r(x_i)}\left|\nabla \left(w_i^n-z_i^n\right)\right|^2{\mathrm {d}}V_g\\&+2\left\Vert\nabla w_i^n\right\Vert_{L^1(\Sigma )}\left\Vert\nabla \left(w_i^n-z_i^n\right)\right\Vert_{L^\infty (B_r(x_i))}\\\le & {} C_r. \end{aligned}$$

Moreover,

$$\begin{aligned} \int _{B_r(x_i)}\widetilde{h}_ie^{a_{ii}\left(w_i^n-\overline{w_i^n}\right)}{\mathrm {d}}V_g\le & {} e^{a_{ii} \left\| w_i^n-\overline{w_i^n}-z_i^n\right\| _{L^\infty \left(B_r\left(x_i\right)\right)}}\int _{B_r(x_i)}\widetilde{h}_ie^{a_{ii}z_i^n}{\mathrm {d}}V_g\\\le & {} C_r\int _{B_r(x_i)}d(\cdot ,x_i)^{2\widetilde{\alpha }_i}e^{a_{ii}z_i^n}{\mathrm {d}}V_g. \end{aligned}$$

Therefore, since \(\widetilde{\alpha }_i\le 0\) and \(a_{ii}\rho _i^n\le 8\pi (1+\widetilde{\alpha }_i)\), we can apply Corollary 2.5 to get the claim:

$$\begin{aligned}&\frac{a_{ii}}{2}\int _{B_r(x_i)}\left|\nabla w_i^n\right|^2{\mathrm {d}}V_g- \rho _i^n\log \int _{B_r(x_i)}\widetilde{h}_ie^{a_{ii}\left(w_i^n-\overline{w_i^n}\right)}{\mathrm {d}}V_g\\&\quad \ge \frac{1}{2a_{ii}}\int _{B_r(x_i)}\left|\nabla \left(a_{ii}z_i^n\right)\right|^2{\mathrm {d}}V_g\\&\qquad -\rho _i^n\log \int _{B_r(x_i)}d(\cdot ,x_i)^{2\widetilde{\alpha }_i}e^{a_{ii}z_i^n}{\mathrm {d}}V_g-C_r\\&\quad \ge -C_r \end{aligned}$$

\(\square \)

Proof of Theorem 1.3

As noticed before, it suffices to prove the boundedness from below of \(J_{\rho ^n}\left(u^n\right)\) for a sequence \(\rho ^n\underset{n\rightarrow +\infty }{\nearrow }\rho ^0\) and a sequence of minimizers \(u^n\) for \(J_{\rho ^n}\). Moreover, due to invariance by addition of constants, one can consider \(v^n\) in place of \(u^n\).

Let us start by estimating the term involving the gradients.

From Corollary 5.3 we deduce that the integral of \(|\nabla w_i^n|^2\) outside a neighborhood of \(x_i\) is uniformly bounded for any \(i\in {\mathcal {I}}\), and the integral on the whole \(\Sigma \) is bounded if \(i\not \in {\mathcal {I}}\).

For the same reason, the integral of \(a_{ij}\nabla w_i^n\cdot \nabla w_j^n\) on the whole surface is uniformly bounded. In fact, if \(a_{ij}\ne 0\), then \(x_i\ne x_j\), then

$$\begin{aligned} \left|\int _\Sigma \nabla w_i^n\cdot \nabla w_j^n{\mathrm {d}}V_g\right|\le & {} \int _{\Sigma {{\setminus }} B_r(x_j)}\left|\nabla w_i^n\cdot \nabla w_j^n\right|{\mathrm {d}}V_g+\int _{\Sigma {{\setminus }} B_r(x_i)}\left|\nabla w_i^n\cdot \nabla w_j^n\right|{\mathrm {d}}V_g\\\le & {} \left\Vert\nabla w_i^n\right\Vert_{L^{q'}(\Sigma )}\left\Vert\nabla w_j^n\right\Vert_{L^{q''}\left(\Sigma {{\setminus }} B_r\{x_j\}\right)}\\&+\left\Vert\nabla w_i^n\right\Vert_{L^{q''}\left(\Sigma {{\setminus }} B_r\{x_i\}\right)}\left\Vert\nabla w_j^n\right\Vert_{L^{q'}(\Sigma )}\\\le & {} C_r, \end{aligned}$$

with q as in Corollary 5.3, \(q'=\left\{ \begin{array}{ll}\frac{2q}{3q-2}<2&{}\text{ if } q<2\\ 1&{}\text{ if } q\ge 2\end{array}\right. \) and \(q''=\left\{ \begin{array}{ll}\frac{2q}{2-q}&{}\text{ if } q<2\\ \infty &{}\text{ if } q\ge 2\end{array}\right. \).

Therefore, we can write

$$\begin{aligned} \sum _{i,j=1}^Na^{ij}\int _\Sigma \nabla v_i^n\cdot \nabla v_j^n{\mathrm {d}}V_g= & {} \sum _{i,j=1}^Na_{ij}\int _\Sigma \nabla w_i^n\cdot \nabla w_j^n{\mathrm {d}}V_g\\\ge & {} \sum _{i\in {\mathcal {I}}}a_{ii}\int _{B_r(x_i)}\left|\nabla w_i^n\right|^2{\mathrm {d}}V_g-C_r. \end{aligned}$$

To deal with the other term in the functional, we use the boundedness of \(w_i^n\) away from \(x_i\): choosing r as in Lemma 5.4, we get

$$\begin{aligned}\int _\Sigma \widetilde{h}_i^ne^{v_i^n-\overline{v_i^n}}{\mathrm {d}}V_g\le & {} 2\int _{B_r(x_i)}\widetilde{h}_i^ne^{v_i^n-\overline{v_i^n}}{\mathrm {d}}V_g\\= & {} 2\int _{B_r(x_i)}\widetilde{h}_ie^{\sum _{j=1}^Na_{ij}\left(w_j^n-\overline{w_j^n}\right)}{\mathrm {d}}V_g\\\le & {} C_r\int _{B_r(x_i)}\widetilde{h}_ie^{a_{ii}\left(w_i^n-\overline{w_i^n}\right)}{\mathrm {d}}V_g. \end{aligned}$$

Therefore, using Lemma 5.4 we obtain

$$\begin{aligned} J_{\rho ^n}\left(v^n\right)= & {} \frac{1}{2}\sum _{i,j=1}^Na^{ij}\int _\Sigma \nabla v_i^n\cdot \nabla v_j^n{\mathrm {d}}V_g-\sum _{i=1}^N\rho _i^n\left(\log \int _\Sigma \widetilde{h}_i^ne^{v_i^n}{\mathrm {d}}V_g-\overline{v_i^n}\right)\\\ge & {} \sum _{i\in {\mathcal {I}}}\left(\frac{a_{ii}}{2}\int _{B_r(x_i)}\left|\nabla w_i^n\right|^2{\mathrm {d}}V_g-\rho _i^n\left(\log \int _{B_r(x_i)}\widetilde{h}_ie^{a_{ii}w_i^n}{\mathrm {d}}V_g-a_{ii}\overline{w_i^n}\right)\right)-C_r\\\ge & {} -C_r \end{aligned}$$

Since the choice of r does not depend on n, the proof is complete. \(\square \)