On the critical points of solutions of PDE: The case of concentrating solutions on the sphere

1.   Introduction

In this paper, we continue the study of the critical points of positive solutions to nonlinear pde's, that we started in ^[1] and ^[2], extending it when the underlying domain is contained in a compact manifold $\mathcal M$. Our aim is to give an estimate on the possible number of maxima, minima, and saddle points of positive solutions to some nonlinear pde's and to their index of critical point. We examine first the case of the sphere in Theorem 1.1, and then we extend the result to a more general surface $\mathcal M$ in Theorem 1.4. We consider the mean field problem in a smooth domain $D\subset \mathcal M$

where $\mathcal M$ is a compact Riemannian surface with metric $g$, $h(x)$ is a $smooth$ function strictly positive on $D$, $\rho$ is a positive constant, $\Delta_g$ is the Laplace Beltrami operator on $\mathcal M$, and $dV_g$ is the volume form on $\mathcal M$.

The Mean Field equation appears in conformal geometry in the problem of understanding the possible Gauss curvatures $h(x)$ of metrics on $\mathcal M$ conformal to the standard metric. When $\mathcal M = S^2$, it is called the Nirenberg problem, and one can see as references the papers ^[3,4,5,6] and references therein. It arises also in some physical models as the mean-field limit of point vortices in the theory of Euler flows, as in ^[7,8,9,10]. And also in the abelian Chern–Simons–Higgs models, see ^{[11,12,13,14,15,16]}.

To state our main result, we need to recall some properties of solutions to Eq (1.1). Let $D\subset \mathcal M$ be a smooth bounded domain. Assume $h$ is a smooth function such that $\inf_{D} h(x) > c$ for some $c > 0$. (By smooth, we mean locally analitic). By ^[17], problem (1.1) has a solution in $H^{1}_0(D)$ for any $\rho\in (8k\pi, 8(k+1)\pi)$ at least when $D$ is not simply connected, while it has a solution in $H^{1}_0(D)$ for any $\rho\in (0, 8\pi)$ for every $D$. The case of $\rho = 8\pi k$ is more complicated and depends on the domain $D$, on the value of $m$ and on the manifold $\mathcal M$. Here we assume to have a sequence of solutions $u_n$ to

such that $\rho_n\to 8\pi m$, for some $m\geq 1$, as $n\to \infty$. It is well known that, see ^[5,17], there exist $m$ points $\{P_1, .., P_m\}\subset D$ and $m$ sequences of points $p_{i, n}\to P_i$ (as $n\to \infty$) for $i = 1, .., m$ such that $u_n(p_{i, n})\to +\infty$ as $n\to \infty$; $u_n-\log \int_D h(x) e^{u_n}dV_g\to -\infty$ uniformly on compact sets of $\overline D\setminus\{P_1, .., P_m\}$ as $n\to \infty$. We say that $u_n$ is a sequence of $m$ blowing-up solutions to Eq (1.2). The value of $8\pi$ comes from the standard bubble solutions to

In fact, in a shrinking ball centered at $p_{i, n}$, it is possible to appropriately rescale the solution $u_n$ to Eq (1.2) and show that this rescaling converges in $C^2_{loc}({{{\rm{I\mskip -3.5mu R}}}}^2)$ to the solution $v$ of Eq (1.3). Thus, each concentration point $P_i$ contributes an amount of $8\pi$.

Our main result is the following:

Theorem 1.1. Let $\mathcal M = \mathcal S^2$ and $D\subset \mathcal S^2$ a smooth domain of Euler characteristic $\chi(D)$. Assume $h$ is a smooth function such that $\inf_{D} h(x) > c$ for some $c > 0$. Assume we have a sequence of solutions $u_n$ to Eq (1.2) such that $\rho_n\to 8\pi m$, for some $m\geq 1$, as $n\to \infty$. Then, $u_n$ is a sequence of $m$ blowing-up solutions, and for $n$ large enough

More precisely, we have that, for $n$ large, there exists exactly one critical point (a nondegenerate maximum) for $u_n$ in $B_{\delta}(P_i)$ $i = 1, .., m$ and ${\delta}$ small. Next, denoting by $D' = D\setminus\cup_{i = 1}^m B_{\delta}(P_i)$ and $\mathcal C_n$ the set of critical points of $u_n$ in $D'$ we have that $u_n$ admits at least $m-\chi(D)$ nondegenerate saddle points in $D'$ and

We can also construct some examples when the estimate in Eq (1.4) is optimal or not.

Corollary 1.2. There exists a domain $D\subset \mathcal S^2$ and a sequence of solutions $u_n$ to Eq (1.2) that blow-up at $m\geq 1$ points $\{P_1, \dots, P_m\}$ such that, for $n$ large enough,

Moreover, all critical points of $u_n$ are nondegenerate; $m$ of them are local maxima and $m-\chi(D)$ saddle points.

Corollary 1.3. There exists a domain $D\subset \mathcal S^2$ and a sequence of solutions $u_n$ to Eq (1.2) that blow-up at $m\geq 1$ points $\{P_1, \dots, P_m\}$ such that, for $n$ large enough

The results of Theorem 1.1 and its corollaries can be generalized to a more general smooth surfaces $\mathcal M$ when the underlying domain $D$ is all contained in a chart of local isotermal coordinates. And this is possible at least when $D$ is sufficiently small with respect to the coordinates. We will say that $D$ is $suitable$ when there exists a system of local isotermal coordinates such that $D$ is contained in a unique chart. For a domain $D$ that is $suitable$, we can prove the following:

Theorem 1.4. Let $\mathcal M$ be a smooth surface and $D\subset \mathcal M$ a $suitable$ smooth domain of Euler characteristic $\chi(D)$. Assume the assumptions of Theorem 1.1 are satisfied. Then the results of Theorem 1, Corollary 1.2, 1.3 hold for the solutions $u_n$ to Eq (1.2) in $D\subset \mathcal M$.

Next, we consider the analog of problem (1.1) in a smooth bounded domain $\Omega\subset {{{\rm{I\mskip -3.5mu R}}}}^2$. Let $\Omega\subset {{{\rm{I\mskip -3.5mu R}}}}^2$ be a smooth bounded domain with $k\geq 0$ holes. Assume $V(x)$ is a smooth function such that $\inf_{\Omega} V(x) > c$ for some $c > 0$. Assume we have a sequence of solutions $\tilde u_n$ to

such that ${{\lambda}}_n\int _{\Omega} V(x) e^{\tilde u_n} dx\to 8\pi m$ as $n\to \infty$, for some integer $m\geq 1$. Then, there exist $m$ points $\{P_1, .., P_m\}\subset \Omega$ and $m$ sequences of points $x_{i, n}\to P_i$ (as $n\to \infty$) for $i = 1, .., m$, such that $\tilde u_n(x_{i, n})\to +\infty$ as $n\to \infty$; $\tilde u_n\to K(x): = 8\pi\sum_{i = 1}^m G(x, P_i)$ uniformly on compact sets of $\overline \Omega\setminus\{P_1, .., P_m\}$ as $n\to \infty$, where $G(x, y)$ is the Green function of the domain $\Omega$ with Dirichlet boundary conditions and pole in $y\in \Omega$. We say that $\tilde u_n$ is a sequence of $m$ blowing-up solutions to Eq (1.7).

In this case, we can prove the following result:

Theorem 1.5. Let $\Omega\subset {{{\rm{I\mskip -3.5mu R}}}}^2$ be a smooth bounded domain with $k\geq 0$ holes. Assume we have a sequence of solutions $\tilde u_n$ to Eq (1.7) such that ${{\lambda}}_n\int _{\Omega} V(x) e^{\tilde u_n} dx\to 8\pi m$ as $n\to \infty$, for some integer $m\geq 1$. Then, when $n$ is large enough,

More precisely, we have that, for $n$ large, there exists exactly one critical point (a nondegenerate maximum) for $\tilde u_n$ in $B_\rho(P_i)$ $i = 1, .., m$ and $\rho$ small. Next, denoting by $\Omega' = {\Omega}\setminus\cup_{i = 1}^m B_\rho(P_i)$ and $\mathcal C_n$ the set of critical points of $\tilde u_n$ in $\Omega'$ we have that $\tilde u_n$ admit at least $m+k-1$ nondegenerate saddle points in $\Omega'$ and

2.   Proofs

In this section, we collect the proofs of the previous results.

Proof of Theorem 1.1. We take a point $N\in {{\mathcal{S}}}^2$, such that $N\notin \bar D$, to be the north pole of the sphere. Then we introduce the standard coordinates on the sphere $\mathcal S^2$. We use the stereographic projection $\psi:\mathcal S^2\setminus\{N\}\to {{{\rm{I\mskip -3.5mu R}}}}^2$. We let $\Omega: = \psi(D)\subset {{{\rm{I\mskip -3.5mu R}}}}^2$ and $v_n(x): = u_n(\psi^{-1}(x))$ for $x\in \Omega$. In these coordinates the functions $v_n$ satisfy

where

is the conformal factor. Next we let

and $V(x): = h(x)e^{\psi(x)}$. Then $v_n$ solves

for some $V(x)$ (which is locally analytic in $\Omega$) and for some $\lambda_n\in (0, \infty)$. Moreover, since $\rho_n\to 8\pi m$ as $n\to \infty$, then

Finally, $\Omega = \psi(D)$ and $\chi(\Omega) = \chi(D)$. Since $\Omega\subset {{{\rm{I\mskip -3.5mu R}}}}^2$ and it is smooth, then $\chi(D) = 1-k$ where $k$ is the number of the holes of $\Omega$. Then, the claim follows from Theorem 1.5.

Proof of Corollaries 1.2 and 1.3. In ^[2], the authors construct a suitable domain $\Omega_1\subset {{{\rm{I\mskip -3.5mu R}}}}^2$ in which $v_n$ (the solution to Eq (2.1), with $V(x) = 1$) has exactly $2m+k-1$ nondegenerate critical points ($m$ maxima and $m+k-1$ saddle points). The very same construction can be done for the case of solutions to Eq (2.1) for a positive, smooth $V(x)$. Pulling back the domain $\Omega$ on the sphere $\mathcal S^2$ gives the desired example on $\mathcal S$.

In Theorem 1.4 in ^[2], an example of a domain $\Omega_2\subset {{{\rm{I\mskip -3.5mu R}}}}^2$ in which $v_n$ has at least $2m+k+1 = 2m-\chi(\Omega_2)+2$ nondegenerate critical points is given. This provides the example in Corollary 1.3.

Proof of Theorem 1.4. We consider the local isotermal coordinates such that $D$ is contained in a unique chart. In these coordinates, Eq (1.2) becomes Eq (2.1), where $e^{\psi(x)}$ is the conformal factor which is locally analitic since we are assuming $\mathcal M$ is smooth. The proof follows as in the case of Theorem 1.1 and its corollaries.

Proof of Theorem 1.5. The proof is similar to the proof of Theorem 1.1 in ^[2]. First, we prove the following statement:

There exists $\rho > 0$ such that $\tilde u_n$ has a unique nondegenerate critical point (the maximum) in $B_\rho(P_i)$ for $i = 1, \dots, m$ when $n$ is large enough.

By contradiction, we assume that there exists $\xi_n\in B_{\rho_n}(x_{i, n})\setminus\{x_{i, n}\}$ such that $\rho_n\to 0$ and $\nabla u_n(\xi_n) = 0$. We have to distinguish two cases: ⅰ) $\xi_n\in B_{R\delta_{i, n}}(x_{i, n})$ for some $i\in \{1, .., m\}$, for some $R > 0$; ⅱ) $\xi_n \notin B_{R\delta_{i, n}}(x_{i, n})$ for any $R > 0$, where $\delta_{i, n}$ satisfies

For $i = 1, \dots, m$, we let

The function $\widehat u_{i, n}(x)$ satisfies

It is standard that

see, ^{[18,19,20,21,22]}. Denote by $\widehat \xi_n: = \frac{\xi_n-x_{i, n}}{{\delta}_{i, n}}$. Then $\nabla \widehat u_{i, n}(\widehat \xi_n) = \nabla \tilde u_n(\xi_n) = 0$ and $|\widehat \xi_n|\leq R$. Up to a subsequence, $\widehat \xi_n\to \widehat \xi$ and by the previous convergence, $\nabla U(\widehat \xi) = 0$. Then the definition of $U(x)$ implies $\widehat \xi = 0$. This is not possible, since $x = 0$ is a nondegenerate maximum point for the function $U(x)$ and the functions $\widehat u_{i, n}$ have a maximum in $x = 0$ for every $n$. This also shows that the point $x_{i, n}$ is a nondegenerate maximum for $\tilde u_n(x)$ and that $index_{x_{i, n}} (\nabla \tilde u_n) = 1$.

Case ⅱ). In this case, we have that $\xi_n\to P_i$. Denoting by $r_n: = |\xi_n-x_{i, n}|$, we have that $\frac{{\delta}_{i, n}}{r_n}\to 0$ as $n\to 0$. We define the function ${\bar {u}}_{i, n}(x): = \tilde u_n(r_nx+x_{i, n})+4 \log r_n$. Green's representation formula gives

First we observe that $I_1 = o(1)$ as $n\to \infty$ since $\tilde u_n(y)$ is bounded in $\Omega\setminus \cup_{i} B_R(x_{i, n})$ and ${{\lambda}}_n\to0$. The second term can be estimated as:

The last term is given by:

In the last line we use that, by ^[5],

Putting together the previous estimates, we have that

We let $\bar \xi_n: = \frac {\xi_n-x_{i, n}}{r_n}$. Then $\nabla \bar u_{i, n}(\bar \xi_n) = \nabla \tilde u_n(\xi_n) = 0$ and $|\bar \xi_n| = 1$. Up to a subsequence, $\bar\xi_n\to \bar \xi$. The previous convergence gives $\nabla V(\bar \xi) = 0$. This is a contradiction. Now we let $\Omega' = \Omega\setminus \cup_{i = 1}^m B_\rho(P_i)$ and we give an estimate on the critical points of $\tilde u_n$ in $\Omega'$. To this end, we observe that a solution $\tilde u_n(x)$ to Eq (1.7) is real analytic in a neighborhood of $x_0$, for every $x_0\in \Omega$, (one can see ^[23]). Moreover it is known that $\tilde u_n(x) \to K(x): = \sum_{i = 1}^m 8\pi G(x, P_i)$ in $\Omega'$. The function $K(x)$ is harmonic and non-trivial in $\Omega'$. Then it has only a finite number of critical points $\{z_1, \dots, z_l\}$, which are saddle points of finite multiplicity $m_j\geq 1$ and $index_{z_j}\left(\nabla K\right)\leq -1$. Whenever $index_{z_j}\left(\nabla K\right) = -1$, then $z_j$ is a nondegenerate saddle point; see Proposition 5.1 in ^[2]. Moreover, we can adapt the proof of Proposition 5.2 in ^[2] getting that $\tilde u_n$, for $n$ large enough, has only a finite number of isolated critical points that we denote by $\{z_{1, n}, \dots, z_{l_n, n}\}$. These points converge to the critical points $\{z_1, \dots, z_l\}$ of $K(x)$. Moreover $index_{z_{j, n}}\left(\nabla u_n\right)\in\{ -1, 0, 1\}$ and, whenever the index is $1$, then $z_{j, n}$ is a nondegenerate maximum, while whenever the index is $-1$, $z_{j, n}$ is a nondegenerate saddle point.

Finally, as in Proposition 5.3 and 5.4 in ^[2], we use the Poincarè Hopf formula with $v = \nabla \tilde u_n$ in $\Omega$ (observe that by Hopf Lemma $\nabla \tilde u_n\cdot \nu < 0$) to have $\sum index_{z_{j, n}}\left(\nabla u_n\right) = \chi(\Omega)$, and by the first assertion, $m+\sum_{\mathcal C_n} index_{z_{j, n}}\left(\nabla u_n\right) = \chi(\Omega) = 1-k$. The previous result on the critical points of $\tilde u_n$ then implies that $\tilde u_n$ has at least $m-k-1$ nondegenerate saddle points (of index $-1$) in $\mathcal C_n$ and concludes the proof.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the e.INS—Ecosystem of Innovation for Next Generation Sardinia (cod. ECS 00000038) and Studio di modelli nelle scienze della vita DM 737/2021 risorse 2022–2023, Uniss (CUP J55F21004240001).

Conflict of interest

The authors declare there is no conflict of interest.