Abstract
We consider the prescribed scalar curvature problem on \( {{\mathbb {S}}}^N \)
under the assumptions that the scalar curvature \({\tilde{K}}\) is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Given the N-th sphere \(({{\mathbb {S}}}^N, g) \) equipped with the standard metric g and a fixed smooth function \({\tilde{K}} \), the prescribed scalar curvature problem on \({{\mathbb {S}}}^N\) consists in understanding whether it is possible to find another metric \( {\tilde{g}} \) in the conformal class of g, such that the scalar curvature of \( {\tilde{g}} \) is \({\tilde{K}} \). For some positive function \(v: {{\mathbb {S}}}^N \rightarrow {\mathbb {R}}\), and a related conformal change of the metric
the scalar curvature with respect to \({\tilde{g}}\) is given by
where \(\Delta _{{{\mathbb {S}}}^N}\) is the Laplace–Beltrami operator on \({{\mathbb {S}}}^N\). Thus the prescribed scalar curvature problem on \({{\mathbb {S}}}^N\) can be addressed by studying the solvability of the problem
Testing the Eq. (1.1) against v and integrating on \({{\mathbb {S}}}^N\), we get that a necessary condition for the solvability of this problem is that \({\tilde{K}}(y) \) must be positive somewhere. There are other obstructions for the existence of solutions, which are said to be of topological type. For instance, a solution v must satisfy the following Kazdan-Warner type condition (see [15]):
This condition is a direct consequence of Theorem 5.17 in [16], where Kazdan and Warner proved that given a positive solution v to
on the standard sphere \({{\mathbb {S}}}^N\), \(N\ge 3\), then
for any spherical harmonics F of degree 1. Taking \(a={N+2 \over N-2}\), \(H= {\tilde{K}}\) and \(F=y\) in (1.3), we can obtain condition (1.2). The problem of determining which \({\tilde{K}}(y) \) admits a solution has been the object of several studies in the past years. We refer the readers to [2,3,4, 6,7,8, 10, 14, 15, 19, 30], and the references therein.
By using the stereo-graphic projection \(\pi _N: {\mathbb {R}}^N \rightarrow {\mathbb {S}}^N{\setminus } \{(0,0,\ldots , 1)\}\), the prescribed scalar curvature problem on \({{\mathbb {S}}}^N\), i.e. (1.1), can be transformed into the following semi-linear elliptic equation
Here \(2^*= {2N \over N-2}\), \(K(y)= {\tilde{K}} (\pi _N y)\), and \(D^{1,2}({\mathbb {R}}^N) \) denote the completion of \(C_c^\infty ({\mathbb {R}}^N) \) with respect to the norm \(\int _{{\mathbb {R}}^N}|\nabla v|^2 \). It is of interest to establish under what kind of assumptions on K problem (1.4) admits one or multiple solutions.
For \(N=3 \), Li [17] showed problem (1.4) has infinitely many solutions provided that K(y) is bounded below, and periodic in one of its variables, and the set \(\{x\, | \, K(x)=\max _{y\in {\mathbb {R}}^3}K(y)\} \) is not empty and contains at least one bounded connected component.
If K has the form \(K(y)=1+\epsilon h(y) \), namely it is a perturbation of the constant 1, Cao et al. [5] proved the existence of multiple solutions.
If K(y) has a sequence of strictly local maximum points moving to infinity, Yan [32] constructed infinitely many solutions.
Wei and Yan [31] showed that problem (1.4) has infinitely many solutions provided K is radially symmetric \(K(y) = K(r)\), \(r= |y|\), and has a local maximum around a given \(r_0 >0\). More precisely, they ask that there are \( r_0 \), \(c_0> 0 \) and \( m \in [2, N-2) \) such that
where \(\sigma , \delta \) are small positive constants. In order to briefly discuss the main results in [31], we will recall the expression of Aubin-Talenti bubbles. It is well known (see [29]) that all solutions to the following problem
are given by
and \(c_N= [N(N-2)]^{\frac{N-2}{4}}\). The solutions in [31] are obtained by gluing together a large number of Aubin-Talenti bubbles, which looks like
where \( {\bar{\Lambda }} \) is a positive constant and the points \(x_j\) are distributed along the vertices of a regular polygon of k edges in the \((y_1,y_2)\)-plane, with \(|x_j| \rightarrow r_0\) as \(k \rightarrow \infty \):
with \( {\tilde{r}} \rightarrow r_0 \) as \( k \rightarrow \infty \).
Under a weaker symmetry condition for \(K(y)=K(|y'|,y'') \) with \(y=(|y'|,y'')\in {\mathbb {R}}^2\times {\mathbb {R}}^{N-2} \), Peng et al. [27] constructed infinitely many bubbling solutions, which concentrate at the saddle points of the potential K(y) . Guo and Li [11] admitted infinitely many solutions for problems (1.4) with polyharmonic operators. For fractional case, we refer to [13, 23].
The study of other aspects of problem (1.4), such as radial symmetry of their solutions, uniqueness of solutions, Liouville type theorem, a priori estimates, and bubbling analysis, have been the object of investigation of several researchers. We refer the readers to the papers [1, 9, 18, 20,21,22, 25, 26, 32] and the references therein.
Recently, Guo et al. [12] investigated the spectral property of the linearized problem associated to (1.4) around the solution \( {\tilde{u}}_k \) found in [31]. They proved a non-degeneracy result for such operator by using a refined version of local Pohozaev identities. As an application of this non-degeneracy result, they built new type of solutions by gluing another large number of bubbles, whose centers lie near the circle \(|y|= r_0 \) in the \((y_3,y_4) \)-plane.
All these results concern solutions made by gluing together Aubin-Talenti bubbles with centres distributed along the vertices of one or more planar polygons, thus of two-dimensional nature. The purpose of this paper is to present a different type of solution to (1.4) with a more complex concentration structure, which cannot be reduced to a two-dimensional one.
To present our result, we assume that K is radially symmetric and satisfies the following condition \(({{\textbf{H}}})\): There are \( r_0 \) and \(c_0> 0 \) such that
where \(\sigma , \delta >0 \) are small constants, and
There is a slight difference between our assumptions on K(s) and the ones in [31]. We will comment on this issue later.
Without loss of generality, we assume \( r_0= 1, \, K(1) =1\). For any integer k, we denote
and set \( u(y)={\textbf{r}}^{-\frac{N-2}{2}}v \big (\frac{|y|}{{\textbf{r}}}\big ) \). Then the problem (1.4) can be rewritten, in terms of u, as
We define
for k integer large, where
Here \( {{\textbf{0}}} \) is the zero vector in \({\mathbb {R}}^{N-3} \) and h, r are positive parameters.
We shall construct a family of solutions to problem (1.8) which are small perturbations of \(W_{r,h,\Lambda }\). More precisely, the Aubin-Talenti bubbles are now centred at points lying on the top and the bottom circles of a cylinder and this configuration is now invariant under a non-trivial sub-group of O(3) rather than O(2).
Throughout of the present paper, we assume \(N\ge 5\) and \((r,h, \Lambda ) \in {{{\mathscr {S}}}_k} \), where
with \( \Lambda _0, B' \) being the constants in (3.7), (3.10) and \( {\hat{\sigma }}\) a fixed small number, independent of k.
Since \(h \rightarrow 0\) as \(k \rightarrow \infty \), then the two circles where the points \({{\overline{x}}}_j\) and \({\underline{x}}_j\) are distributed become closer to each other as k increases.
In this paper, we shall prove that for any k large enough, problem (1.8) admits a family of solutions \(u_k\) with the approximate form
Moreover, these solutions are polygonal symmetry in the \((y_1,y_2) \)-plane, even in the \(y_3 \) direction and radially symmetric in the variables \(y_4,\ldots ,y_N \). Our solutions are thus different from the ones obtained in [31] and have strong analogies with the doubling construction of the entire finite energy sign-changing solutions for the Yamabe equation in [24].
Define the symmetric Sobolev space:
where \( \theta = \arctan {\frac{y_2}{y_1}} \).
Let us define the following norms which capture the decay property of functions
and
where
for some \(\epsilon _1\) small. The main results of this paper are the following:
Theorem 1.1
Let \(N\ge 5\) and suppose that K(|y|) satisfies \(({{\textbf{H}}})\). Then there exists a large integer \(k_0\), such that for each integer \(k\ge k_0\), problem (1.8) has a solution \(u_k\) of the form
where \( \phi _k \in H_s, \, (r_k,h_k, \Lambda _k) \in {{\mathscr {S}}_k}\), and \( \phi _k \) satisfies
Equivalently, problem (1.4) has solution \(v_k(y) \) of the form
with \(\textbf{r}\) as in (1.7).
Let us sketch the proof of Theorem 1.1. The first step in our argument is to find \(\phi \) so that \(u=W_{r,h,\Lambda }+\phi \) solves the auxiliary problem
for some constants \( c_\ell \) for \( \ell = 1,2,3 \). In (1.16), the functions \(\overline{{\mathbb {Z}}}_{\ell j}\) and \(\underline{{\mathbb {Z}}}_{\ell j}\) are given by
for \( j= 1, \ldots , k \). Moreover, the function \(\phi \) belongs to the set \({\mathbb {E}}\) given by
From the linear theory developed in Sect. 2, problem (1.16) can be solved by means of the contraction mapping theorem. More precisely, we prove that, for any \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\) there exist \(\phi = \phi _{r,h,\Lambda } \in {\mathbb {E}} \) and constants \(c_\ell \), \(\ell =1,2,3\) which solve the auxiliary problem (1.16).
After the correction \(\phi \) has been found, we shall choose \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k} \) so that the multipliers \(c_\ell =0\,(\ell = 1,2,3) \) in (1.16). As a consequence, we can derive the results as in Theorem 1.1.
Equation (1.8) is the Euler–Lagrange equation associated to the energy functional
Thus, roughly speaking, if \((r,h,\Lambda )\) is a critical point of function
then the constants \( c_ \ell ,\,\ell =1,2,3 \) would be zero. Thus finding solutions of problem (1.8) would be reduced to find a critical point of \( F(r,h,\Lambda ) \). This is the result in Proposition 3.1.
An important work of this paper is to give an accurate expression of \( F(r,h,\Lambda ) \) (see Proposition 3.2). Under the assumptions \( r \sim k^{\frac{N-2}{N-2-m}}, h \rightarrow 0, \frac{1}{hk} \rightarrow 0 \) as \(k \rightarrow \infty \), we first get the expansion of the energy functional \( I(W_{r,h,\Lambda } )\)
where \( A_i\) for \(i=1,2,3\) and \( B_j\) for \( j=4,5 \) are constants. We denote
and let h be the solution of \( \partial _h {\mathcal {G}}(h)= 0\), then
for some \(B' >0 \). If \( r \sim k^{\frac{N-2}{N-2-m}}, h \sim \frac{B'}{k^{\frac{N-3}{N-1}}} \), then
for some constant \({\tilde{B}}\).
However, we now find that the term \(O \Big (\frac{1}{k^{\frac{(N-2)m}{N-2-m}+\sigma }}\Big ) \) in the expansion of \(F_1 (r,h, \Lambda )\) competes with the term \(\frac{B_5 k}{r^{N-2} h^{N-3} \sqrt{1-h^2}}\). This makes it impossible to identify a critical point for \( F_1 (r,h,\Lambda )\). In reality, though the remainder \(O \Big (\frac{1}{k^{\frac{(N-2)m}{N-2-m}+\sigma }}\Big ) \) can be estimated in a more accurate way (see Proposition A.4) under our assumption \(({\textbf{H}}) \).
We need to expand the full energy \( F(r,h,\Lambda ) = I(W_{r, h, \Lambda }+\phi _{r, h, \Lambda })\). We need a strong control on the size of \(\phi _{r, h, \Lambda }\) in order not to destroy the critical point structure of \(F_1 (r,h, \Lambda )\) and to ensure the qualitative properties of the solutions as stated in Theorem 1.1. This is another delicate step of our construction, where we make full use of the assumption \(({\textbf{H}})\) on K.
Structure of the paper. The remaining part of this paper is devoted to the proof of Theorem 1.1, which will be organized as follows:
-
1.
In Sect. 2, we will establish the linearized theory for the linearized projected problem. We will give estimates for the error terms in this Section.
-
2.
In Sect. 3, we shall prove Theorem 1.1 by showing there exists a critical point of reduction function \( F(r,h,\Lambda ) \).
-
3.
Some tedious computations and some useful Lemmas will be given in Appendices 3–4.
Notation and preliminary results. For the readers’ convenience, we will provide a collection of notation. Throughout this paper, we employ \(C, C_j \) to denote certain constants and \( \sigma , \tau , \sigma _j \) to denote some small constants or functions. We also note that \( \delta _{ij} \) is Kronecker delta function:
Furthermore, we also employ the common notation by writing \(O(f(r,h)), o(f(r,h)) \) for the functions which satisfy
and
2 Finite dimensional reduction
For \(j= 1,\ldots , k \), we divide \({\mathbb {R}}^N\) into k parts:
where \( \langle , \rangle _{{\mathbb {R}}^2}\) denote the dot product in \({\mathbb {R}}^2\). For \(\Omega _j\), we further divide it into two parts:
We can know that
and
We consider the following linearized problem
for some constants \(c_{\ell } \).
Coming back to Eq. (1.5), we recall that the functions
belong to the null space of the linearized problem associated to (1.5) around an Aubin-Talenti bubble, namely they solve
It is known [28] that these functions span the set of the solutions to (2.3). This fact will be used in the following crucial lemma which concerns the linearized problem (2.1).
Lemma 2.1
Suppose that \(\phi _{k}\) solves (2.1) for \(f=f_{k}\). If \(\Vert f_{k}\Vert _{**}\) tends to zero as k tends to infinity, so does \(\Vert \phi _{k}\Vert _{*}\).
The norms \(\Vert \cdot \Vert _*\) and \(\Vert \cdot \Vert _{**}\) are defined respectively in (1.12) and (1.13).
Proof
We prove the Lemma by contradiction. Suppose that there exists a sequence of \((r_k, h_k, \Lambda _k)\in {{{\mathscr {S}}}_k}\), and for \(\phi _k\) satisfies (2.1) with \(f=f_k, r= r_k, h= h_k, \Lambda = \Lambda _k\), with \(\Vert f _{k}\Vert _{**}\rightarrow 0 \), and \(\Vert \phi _k\Vert _*\ge c'>0\). Without loss of generality, we can assume that \(\Vert \phi _k\Vert _*=1 \). For convenience, we drop the subscript k.
From (2.1), we know that
For the first term \(M_1 \), we make use of Lemma B.5, so that
For the second term \(M_2 \), we make use of Lemma B.4, so that
In order to estimate the term \(M_3 \), we will first give the estimates of \( \overline{{\mathbb {Z}}}_{1j} \) and \( \underline{{\mathbb {Z}}}_{1j} \)
Combining estimates (2.4) and Lemma B.4, we have
where \(\delta _{\ell 2} =0\) if \(\ell \not = 2\), \(\delta _{\ell 2}=1\) if \(\ell =2\). Similarly, we have
Next, we will give the estimates of \({c_\ell }, \ell = 1,2,3 \). Multiply both sides of (2.1) by \(\overline{{\mathbb {Z}}}_{q 1}, q=1,2,3 \), then we obtain that
Using Lemma B.3, we can get
The discussion on the left side of (2.5) may be more tricky, in fact, we have
Using the property of K(s) , similar to the proof of Lemma B.5, we can get
For \(J_2\), it is easy to derive that
Then, we get
On the other hand, there holds
Note that
for some constant \({\bar{c}}_q >0 \). Then we can get
Then we have
Combining this fact and \(\Vert {\phi }\Vert _{*}= 1 \), we have the following claim:
Claim 1: There exist some positive constants \( {\bar{R}}, \delta _1 \) such that
for some \( l\in \{1,2, \ldots , k\} \).
Since \( \phi \in H_s \), we assume that \(l=1 \). By using local elliptic estimates and (2.7), we can get, up to subsequence, \({{\tilde{\phi }}}(y) = \phi (y-{\overline{x}}_1) \) converge uniformly in any compact set to a solution
for some \(\Lambda \in [L_1,L_2] \). Since \( \phi \) is even in \(y_d, d= 2, 4, \ldots ,N \), we know that u is also even in \(y_d, d= 2, 4, \ldots ,N \). Then we know that u must be a linear combination of the functions
From the assumptions
we can get
and
By taking limit, we have
So we have \(u=0 \). This is a contradiction to (2.8). \(\square \)
For the linearized problem (2.1), we have the following existence, uniqueness results. Furthermore, we can give the estimates of \( \phi \) and \(c_\ell , \ell =1,2,3 \).
Proposition 2.2
There exist \(k_0>0 \) and a constant \(C>0 \) such that for all \(k\ge k_0 \) and all \(f\in L^{\infty }({\mathbb {R}}^N) \), problem (2.1) has a unique solution \(\phi \equiv {\textbf{L}}_k(f) \). Besides,
Proof
Recall the definition of \({\mathbb {E}}\) as in (1.17), we can rewrite problem (2.1) in the form
in the sense of distribution. Furthermore, by using Riesz’s representation theorem, Eq. (2.10) can be rewritten in the operational form
where \({\mathbb {I}}\) is identity operator and \({\mathbb {T}}_k\) is a compact operator. Fredholm’s alternative yields that problem (2.11) is uniquely solvable for any \( {\tilde{f}} \) when the homogeneous equation
has only the trivial solution. Moreover, problem (2.12) can be rewritten as following
Suppose that (2.13) has nontrivial solution \(\phi _k\) and satisfies \(\Vert \phi _k\Vert _{*}=1\). From Lemma 2.1, we know \(\Vert \phi _k\Vert _{*}\) tends to zero as \( k\rightarrow +\infty \), which is a contradiction. Thus problem (2.12) (or (2.13)) only has trivial solution. So we can get unique solvability for problem (2.1). Using Lemma 2.1, the estimates (2.9) can be proved by a standard method. \(\square \)
We can rewrite problem (1.16) as following
where
and
Next, we will use the Contraction Mapping Principle to show that problem (2.14) has a unique solution in the set that \(\Vert \phi \Vert _* \) is small enough. Before that, we will give the estimate of \({{\textbf{N}}}(\phi ) \) and \( {\textbf{l}}_k \).
Lemma 2.3
Suppose \( N\ge 5 \). There exists \(C>0\) such that
for all \(\phi \in {\mathbb {E}}\).
Proof
The proof is similar to that of Lemma 2.4 in [31]. Here we omit it. \(\square \)
We next give the estimate of \({\textbf{l}}_k\).
Lemma 2.4
Suppose K(|y|) satisfies \(({\textbf{H}})\) and \(N\ge 5\), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). There exists \(k_0 \) and \(C>0\) such that for all \(k \ge k_0\)
where \(\epsilon _1\) is small constant given in (1.14).
Proof
We can rewrite \({\textbf{l}}_{k}\) as
Assume that \( y \in \Omega _1^+ \), then we get
Thus, we have
We first consider the case \(N=5\). It is easy to get that
When \(N\ge 6\), similar to the proof of Lemma B.1, for any \( 1< \alpha _1 < N-2 \), we have
Since \( \tau \in ( \frac{N-2-m}{N-2}, \frac{N-2-m}{N-2}+ \epsilon _1 ) \), we can choose \(\alpha _1\) satisfies
Then
Then combining (2.16) and (2.17), we can get
For \(S_{12}\), we can rewrite it as following
Similarly to (2.16), we can obtain
For \(N\ge 6\) and the same \(\alpha _1\) as in (2.18), it is easy to derive that
where we have used the fact \( h r > C \frac{{\textbf{r}}}{k}\). Thus, we can obtain that
Next, we consider \(S_{13}\). For \( y \in \Omega _1^+\),
Thus we have
Since \(\big (\frac{N+2}{2}-\frac{N-2-m}{N-2}- \epsilon _1\big )\big |_{N=5} >3\) for \(m\in [2, 3)\), then we have
Combining (2.18), (2.19), (2.20), we obtain
We now consider the estimate of \(S_2\). For \( y\in \Omega _1^+ \), we have
-
If \(|\frac{|y|}{{\textbf{r}}}-1|\ge \delta _1, \) where \( \delta> \delta _1> 0 \), then
$$\begin{aligned} |y-{\overline{x}}_1| \ge \big ||y |-{\textbf{r}}\big |\,-\,\big |{\textbf{r}} -|{\overline{x}}_1|\big | \ge \frac{1}{2} \delta _1 {\textbf{r}}. \end{aligned}$$
As a result, we get
-
If \(|\frac{|y|}{{\textbf{r}}}-1|\le \delta _1,\) then
$$\begin{aligned} \Big [K\Big (\frac{|y|}{{\textbf{r}}}\Big )-1\Big ]&\le C\Big |\frac{|y|}{{\textbf{r}}}-1\Big |^{m} = \frac{C}{{\textbf{r}}}^{{m}}||y|-{\textbf{r}}|^{m}\\&\le \frac{C}{{\textbf{r}}}^{m}\Big [\big ||y|-|{\overline{x}}_1|\big |^{m} \,+\,\big ||{\overline{x}}_1|-{\textbf{r}}\big |^{m}\Big ]\\&\le \frac{C}{{\textbf{r}}}^{m}\Big [\big ||y|-|{\overline{x}}_1|\big |^{m} \,+\,\frac{1}{k^{{{\bar{\theta }} m}}}\Big ]. \end{aligned}$$
Thus, we can get, if \(m>3\),
the last inequality holds due to \(\frac{N+2}{2}- \tau -\frac{m+3}{2}>0.\)
On the other hand, if \(m\le 3\), we have
since \(\frac{N+2}{2}- \tau -m>0.\) Thus we have
As a result,
Since \( y \in \Omega _1^+ \), then for \(j= 2, \ldots , k \), there holds
Therefore, it is easy to derive that
Combining (2.22) with (2.23), we obtain
If \(N=5\), we can check that \( \frac{1}{{\textbf{r}}}^{ m}\,=\,\Big (\frac{k}{{\textbf{r}}} \Big )^3 \). Thus, we can rewrite (2.21) as
Therefore, we showed (2.15). \(\square \)
The solvability theory for the projected problem (2.14) can be provided in the following:
Proposition 2.5
Suppose that K(|y|) satisfies \(({\textbf{H}})\) and \(N\ge 5\), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). There exists an integer \(k_0\) large enough, such that for all \(k \ge k_0\) problem (2.14) has a unique solution \(\phi _k\) which satisfies
and
Proof
We first denote
From Proposition 2.2, we know that problem (2.14) is equivalent to the following fixed point problem
where \({\textbf{L}}_k\) is the linear bounded operator defined in Proposition 2.2.
From Lemmas 2.3 and 2.4, we know, for \(\phi \in {\mathcal {B}} \)
So the operator \({\textbf{A}}\) maps from \({\mathcal {B}}\) to \({\mathcal {B}}\). Furthermore, we can show that \({\textbf{A}}\) is a contraction mapping. In fact, for any \(\phi _1, \phi _2 \in {\mathcal {B}}\), we have
Since \({{\textbf{N}}}(\phi )\) has a power-like behavior with power greater than one, then we can easily get
A direct application of the contraction mapping principle yields that problem (2.14) has a unique solution \( \phi \in {\mathcal {B}}\). The estimates for \( c_{\ell }, \ell =1,2,3 \) can be got easily from (2.6). \(\square \)
3 Proof of Theorem 1.1
Proposition 3.1
Let \( \phi _{r,h, \Lambda }\) be a function obtained in Proposition 2.5 and
If \((r,h, \Lambda )\) is a critical point of \(F(r,h,\Lambda )\), then
is a critical point of I(u) in \(H^1({\mathbb {R}}^N)\). \(\square \)
We will give the expression of \(F(r,h,\Lambda )\). We first note that we employ the notation \({\mathcal {C}}(r, \Lambda )\) to denote functions which are independent of h and uniformly bounded.
Proposition 3.2
Suppose that K(|y|) satisfies \(({\textbf{H}})\) and \(N\ge 5 \), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). We have the following expansion as \(k \rightarrow \infty \)
where \(A_1, A_2, A_3, B_4, B_5\) are positive constants.
Proof
The proof of Proposition 3.2 is similar to that of Proposition 3.1 in [31]. We omit it here. \(\square \)
Next, we will give the expansions of \(\frac{\partial F(r,h, \Lambda )}{\partial \Lambda }\) and \( \frac{\partial F(r,h, \Lambda )}{\partial h} \).
Proposition 3.3
Suppose that K(|y|) satisfies \(({\textbf{H}})\) and \(N\ge 5 \), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). We have the following expansion for \(k \rightarrow \infty \)
where \(A_2, A_3, B_4, B_5\) are positive constants.
Proof
The proof of this proposition can be found in [31]. We omit it here.
\(\square \)
Proposition 3.4
Suppose that K(|y|) satisfies \(({\textbf{H}})\) and \(N\ge 5 \), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). We have the following expansion
where \( B_4, B_5\) are positive constants.
Proof
Notice that \(F(r,h, \Lambda )\,=\, I( W_{r, h, \Lambda }+\phi _{r,h, \Lambda })\), there holds
Since \( \int _{{{\mathbb {R}}}^N} U_{{\overline{x}}_j, \Lambda }^{2^*-2} \overline{{\mathbb {Z}}}_{\ell j} \phi _{r,h, \Lambda }= \int _{{{\mathbb {R}}}^N} U_{{\underline{x}}_j, \Lambda }^{2^*-2} \underline{{\mathbb {Z}}}_{\ell j} \phi _{r,h, \Lambda }=0\), we can get easily
Then
where we used the estimates (2.24)-(2.25) and the inequalities
On the other hand, we have
For the second term in (3.5), using the decay property of K(|y|) and orthogonality of \(\phi _{r, h, \Lambda }\), we can show this term is small. In fact, we have
According to the expression of \(W_{r, h, \Lambda }\), we can obtain that
And it’s easy to show that
Combining all above, we can get
Combing (3.6), Proposition A.6 and Lemma B.6, we can get (3.2) \(\square \)
Remark 3.5
The expansions of \(\frac{\partial F(r,h, \Lambda )}{\partial h}\) and \(\frac{\partial F(r,h, \Lambda )}{\partial \Lambda }\) would be applied in the proof of Proposition 3.6, which is essential for proving the existence critical point of \(F(r,h, \Lambda )\). In order to get a proper expansion of \(\frac{\partial F(r,h, \Lambda )}{\partial h}\), we need accurate estimates for \(\phi _{r,h, \Lambda } \). \(\square \)
Rewritten the expansion of the energy functional.
Let \(\Lambda _0\) be
Then it solves
Denote
then
and
and
Let \( {\textbf{h}}\) be a solution of
then
Define
for \({\bar{\theta }}\) is a small constant such that \({\bar{\theta }} \le \frac{\sigma }{100}\). In fact, \( {{\textbf{S}}_k}\) is a subset of \({{{\mathscr {S}}}_k}\). We will find a critical point of \(F(r, h, \Lambda )\) in \({{\textbf{S}}}_k\).
A direct Taylor expansion gives that
where
and
Since \({\mathcal {G}}({\textbf{h}}), {\mathcal {G}} '' ({\textbf{h}}) \) are independent of \(h, r, \Lambda \), for simplicity, in the following, we will denote
Then combining (3.11), (3.12), (3.13), we can get
Therefore, we get
where
Since
then
We now rewrite
Then we can express \( F(r,h, \Lambda )\) as
And similarly, we have
and from (A.28), by using some calculations, we have
Now define
and
where \(\eta _1>0 \) small. We also define the energy level set
We consider the following gradient flow system
The next proposition would play an important role in the proof of Theorem 1.1.
Proposition 3.6
The flow would not leave \( {{\textbf{S}}}_k \) before it reaches \( {{\bar{F}}}^{{\textbf{t}}}_1. \)
Proof
There are three positions that the flow tends to leave \( {\textbf{S}}_k \):
position 1. \(|r-{\textbf{r}}|= \frac{1}{k^{{\bar{\theta }}}} \) and \(|1-{\textbf{h}}^{-1} h| \le \frac{1}{k^{{\bar{\theta }}}}, \quad |\Lambda -\Lambda _0|\le \frac{1}{k^{\frac{3 {\bar{\theta }}}{2}}} \);
position 2. \(|1-{\textbf{h}}^{-1} h|= \frac{1}{k^{{\bar{\theta }}}} \) when \(|r-{\textbf{r}}| \le \frac{1}{k^{{\bar{\theta }}}}, \quad |\Lambda -\Lambda _0|\le \frac{1}{k^{\frac{3 {\bar{\theta }}}{2}}} \);
position 3. \(| \Lambda -\Lambda _0|= \frac{1}{k^{\frac{3 {\bar{\theta }}}{2}}} \) when \(|r-{\textbf{r}}| \le \frac{1}{k^{{\bar{\theta }}}}, \quad |1-{\textbf{h}}^{-1} h| \le \frac{1}{k^{{\bar{\theta }}}}. \)
\(\spadesuit \) We now consider position 1. Since \(|\Lambda -\Lambda _0|\le \frac{1}{k^{\frac{3 {\bar{\theta }}}{2}}} \), it is easy to derive that
Combining (3.15), (3.17), (3.18), we can obtain that, if \((r,h,\Lambda ) \) lies in position 1,
\(\spadesuit \) On the other hand, we claim that it’s impossible for the flow \(\big (r(t), h(t), \Lambda (t)\big )\) leaves \({{\textbf{S}}}_k\) when it lies in position 2. If \(1-{\textbf{h}}^{-1} h= \frac{1}{k^{{\bar{\theta }}}} \), then from (3.16) and (3.17), we have
On the other hand, if \(1-{\textbf{h}}^{-1} h= -\frac{1}{k^{{\bar{\theta }}}} \)
So it’s impossible for the flow leaves \({{\textbf{S}}}_k\) when it lies in position 2.
\(\spadesuit \) Finally, we consider position 3. If \(\Lambda =\Lambda _0+\frac{1}{k^{\frac{3 {\bar{\theta }}}{2}}} \), from (3.1) and (3.17), there exists a constant \( C_1 \) such that
On the other hand, if \(\Lambda =\Lambda _0-\frac{1}{k^{\frac{3 \bar{\theta }}{2}}}\), there exists a constant \(C_2\) such that
Hence the flow \( \big (r(t), h(t), \Lambda (t)\big )\) does not leave \({{\textbf{S}}}_k\) when \(| \Lambda -\Lambda _0|= \frac{1}{k^{\frac{3 {\bar{\theta }}}{2}}}\).
Combining above results, we conclude that the flow would not leave \({{\textbf{S}}}_k\) before it reach \( {\bar{F}}^{{\textbf{t}}}_1\). \(\square \)
Now we give the proof of Theorem 1.1.
Proof of Theorem 1.1
: According to Proposition 3.1, in order to show Theorem 1.1, we only need to show that function \({{\bar{F}}}(r,h, \Lambda )\), and thus \(F(r,h, \Lambda )\), has a critical point in \({{\textbf{S}}}_k\).
Define
Let
We claim that \( {\textbf{c}} \) is a critical value of \( {{\bar{F}}}(r,h, \Lambda ) \) and can be achieved by some \((r,h, \Lambda ) \in {\textbf{S}}_k \). By the minimax theory, we need to show that
-
(i)
\( {\textbf{t}}_1< {\textbf{c}} < {\textbf{t}}_2 \);
-
(ii)
\(\sup _{|r-{\textbf{r}}|= \frac{1}{k^{{\bar{\theta }}}}} {{{\bar{F}}}}\big (\gamma (r,h,\Lambda )\big )<{\textbf{t}}_1,\;\forall \; \gamma \in \Gamma . \)
Using the results in Proposition 3.6 we can prove (i) and (ii) easily.
Finally, for every k large enough, we get the critical point \((r_k, h_k, \Lambda _k) \) of \( F(r,h,\Lambda ) \). \(\square \)
References
Ambrosetti, A., Azorero, G., Peral, J.: Perturbation of \(-\Delta u- u^{\frac{N+2}{N-2}}= 0 \), the scalar curvature problem in \({\mathbb{R}}^{N} \) and related topics. J. Funct. Anal. 165, 117–149 (1999)
Bahri, A., Coron, J.: The scalar-curvature problem on the standard three-dimensional sphere. J. Funct. Anal. 95, 106–172 (1991)
Bianchi, G.: Non-existence and symmetry of solutions to the scalar curvature equation. Commun. Partial Differ. Equ. 21, 229–234 (1996)
Brezis, H., Peletier, L.A.: Elliptic equations with critical exponent on spherical caps of \(S^3 \). J. Anal. Math. 98, 279–316 (2006)
Cao, D., Noussair, E., Yan, S.: On the scalar curvature equation \(-\Delta u=(1+\epsilon K)u^{\frac{N+2}{N-2}} \) in \({\mathbb{R}}^{N}, \) Calc. Var. Partial Differ. Equ. 15, 403–419 (2002)
Chang, S.-Y.A., Yang, P.C.: A perturbation result in prescribing scalar curvature on \(S^{N}\). Duke Math. J. 64, 27–69 (1991)
Chen, C.-C., Lin, C.-S.: Estimate of the conformal scalar curvature equation via the method of moving planes. II. J. Differ. Geom. 49, 115–178 (1998)
Chen, W.-X., Ding, W.-Y.: Scalar curvature on \(S^2 \). Trans. Am. Math. Soc. 303, 369–382 (1987)
Deng, Y., Lin, C.-S., Yan, S.: On the prescribed scalar curvature problem in \({{\mathbb{R} }}^N \), local uniqueness and periodicity. J. Math. Pures Appl. 104, 1013–1044 (2015)
Druet, O.: From one bubble to several bubbles: the low-dimensional case. J. Differ. Geom. 63, 399–473 (2003)
Guo, Y., Li, B.: Infinitely many solutions for the prescribed curvature problem of polyharmonic operator. Calc. Var. Partial Differ. Equ. 46, 809–836 (2013)
Guo, Y., Musso, M., Peng, S., Yan, S.: Non-degeneracy of multi-bubbling solutions for the prescribed scalar curvature equations and applications. J. Funct. Anal. 279, 108553 (2020)
Guo, Y., Nie, J.: Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete Contin. Dyn. Syst. 36, 6873–6898 (2016)
Han, Z.-C.: Prescribing Gaussian curvature on \(S^{2}\). Duke Math. J. 61, 679–703 (1990)
Kazdan, J., Warner, F.: Existence and conformal deformation of metrics with prescribed Gaussian and Scalar curvature. Ann. Math. 101, 317–331 (1975)
Kazdan, J., Warner, F.: Scalar curvature and conformal deformation of Riemannian structure. J. Differ. Geom. 10(1), 113–134 (1975)
Li, Y.Y.: On \(-\Delta u=K(x)u^5 \) in \({{\mathbb{R}}}^3 \). Commun. Pure Appl. Math. 46, 303–340 (1993)
Li, Y.Y.: Prescribing scalar curvature on \(S^3, S^4 \) and related problems. J. Funct. Anal. 118, 43–118 (1993)
Li, Y.Y.: Prescribing scalar curvature on \(S^{n} \) and related problems. I. J. Differ. Equ. 120, 319–410 (1995)
Li, Y.Y.: Prescribing scalar curvature on \(S^{N} \) and related problems. II, Existence and compactness. Commun. Pure Appl. Math. 49, 541–597 (1996)
Li, Y.Y., Wei, J., Xu, H.: Multi-bump solutions of \(-\Delta u=K(x)u^{\frac{n+2}{n-2}} \) on lattices in \({\mathbb{R}}^n \). J. Reine Angew. Math. 743, 163–211 (2018)
Lin, C.S., Lin, S.-S.: Positive radial solutions for \(\Delta u+K(r)u^{\frac{N+2}{N-2}}=0 \) in \({\mathbb{R}}^{n} \) and related topics. Appl. Anal. 38, 121–159 (1990)
Long, W., Peng, S., Yang, J.: Infinitely many positive and sign-changing scalar field equations. Discrete Contin. Dyn. Syst. 36, 917–939 (2016)
Medina, M., Musso, M.: Doubling nodal solution to the Yamabe equation in \(\mathbb{R} ^n \) with maximal rank. J. Math. Pures Appl. (9) 152, 145–188 (2021)
Ni, W.M.: On the elliptic equation \(\Delta u+K(x)u^{\frac{n+2}{n-2}}=0 \) its generalizations and applications in geometry. Indiana Univ. Math. J. 31, 493–529 (1982)
Noussair, E., Yan, S.: The scalar curvature equation on \({\mathbb{R} }^{N}\). Nonlinear Anal. 45, 483–514 (2001)
Peng, S., Wang, C., Wei, S.: Construction of solutions for the prescribed scalar curvature problem via local Pohozaev identities. J. Differ. Equ. 267, 2503–2530 (2019)
Rey, O.: The role of the Green’s function in a nonlinear elliptic problem involving the critical Sobolev exponent. J. Funct. Anal. 89, 1–52 (1990)
Talenti, G.: Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)
Schoen, R., Zhang, D.: Prescribed scalar curvature on the \(n \)-sphere. Calc. Var. Partial Differ. Equ. 4, 1–25 (1996)
Wei, J., Yan, S.: Infinitely many solutions for the prescribed scalar curvature problem on \({{\mathbb{S} }}^N \). J. Funct. Anal. 258, 3048–3081 (2010)
Yan, S.: Concentration of solutions for the scalar curvature equation on \({\mathbb{R}}^{N}\). J. Differ. Equ. 163, 239–264 (2000)
Acknowledgements
L. Duan was supported by the China Scholarship Council and NSFC grant (No.11771167, No.12201140), Technology Foundation of Guizhou Province ([2001]ZK008) and Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515111131). M. Musso was supported by EPSRC research Grant EP/T008458/1. S. Wei was supported by the NSFC Grant (No.12001203) and Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515110622). Some part of the work was done during the visit of L. Duan to Prof. M. Musso at the University of Bath. L. Duan would like to thank the Department of Mathematical Sciences for its warm hospitality and supports.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Expansions for the energy functional
This section is devoted to the computation of the expansion for the energy functional \(I(W_{r,h,\Lambda }) \). We first give the following Lemma.
Lemma A.1
\(N\ge 5 \) and \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). We have the following expansions for \(k \rightarrow \infty \):
where
and
Proof
In fact, for \( \frac{1}{2} < c_3 \le c_4\le 1 \), we have
Without loss of generality, we can assume k is even. It is easy to derive that
Direct computations show that
where \( D_{1}= \frac{1}{{2\pi }^{N-2}} \sum _{i=1}^{\infty } \frac{1}{i^{N-2}} \) and \(\sigma _1(k) \) is defined in (A.4). Using symmetry of function \(\sin x \), we can easily show
Thus we proved (A.1).
Similarly, we can obtain
Consider \( O \big ((h k)^{-1}\big )= o(1) ~\text {as}~ k \rightarrow \infty \). Since
then we have
Combining above calculations, we can obtain that
where \( B_2 \) and \( \sigma _2 \) are defined in (A.3), (A.4). \(\square \)
Lemma A.2
We have the expansion, for \(k \rightarrow \infty \)
and
where \(B_0= \int _{{\mathbb {R}}^N} \frac{1}{(1+z^{2})^{\frac{N+2}{2}}} \) and \( \epsilon _0 \) is constant small enough.
Proof
Let \( {\overline{d}}_j =|{\overline{x}}_1-{\overline{x}}_j|, ~ {\underline{d}}_j =|{\overline{x}}_1-{\underline{x}}_j| \) for \(j=1, \ldots , k \). We consider
First, we have
It is easy to check that
and
Standard calculation implies that
where \(B_0= \int _{{\mathbb {R}}^N} \frac{1}{(1+z^{2})^{\frac{N+2}{2}}} \).
When \( y \in {\mathbb {R}}^N {\setminus } {B_{\frac{{\overline{d}}_i}{4}}({\overline{x}}_1)} \), there holds
It’s easy to get
Combining (A.6), (A.11) and (A.12), we can get
Similarly, we can get
\(\text {for} ~ i=1,\ldots , k. \) \(\square \)
Lemma A.3
Suppose that K(|y|) satisfies \(({\textbf{H}}) \) and \(N\ge 5 \), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). We have the expansion for \(k \rightarrow \infty \)
where \({\mathcal {C}}(r, \Lambda )\) denotes function independent of h and should be order of O(1),
and \( \epsilon _0 \) is constant can be chosen small enough.
Proof
Recalling the definition of I(u) as in (1.18), then we obtain that
According to the expression of \(W_{r,h,\Lambda }\), we have
For \(I_2\), using the symmetry of function \(W_{r,h,\Lambda }\), we have
For \( y\in \Omega _1^{+} \), from Lemma B.1, we have
with \( \epsilon _0 > 0 \) can be chosen small enough. Then we can get
For \(I_{21} \), we can rewrite it as following
Furthermore, we obtain
When \(|\frac{|y|}{{\textbf{r}}}-1|\ge \delta \), there holds
Thus we can easily get
If \(|\frac{|y|}{{\textbf{r}}}-1|\le \delta \), recalling the decay property of K, we can obtain that
Furthermore, recalling \(|{\overline{x}}_1|= r \) and using the symmetry property, we have
where \( e_1=(1, 0,\ldots , 0) \).
We get
here \({\mathcal {C}}(r, \Lambda )\) denote functions which are independent of h and can be absorbed in O(1) .
Similarly, we can also have the following expression
Then, we can obtain that
Finally, we consider \(I_{22} \)
For \(I_{222} \), it is easy to derive that
Moreover, we know that
where \( {\overline{d}}_j =|{\overline{x}}_1-{\overline{x}}_j|\) for \(j=2, \ldots , k\) and \( {\overline{d}}_ 2=|{\overline{x}}_1-{\overline{x}}_2|= 2r \sqrt{1-h^2} \sin {\frac{\pi }{k}}= O\big (\frac{r}{k}\big ) \). Then we get
Next, we consider the term \(I_{223}\). In fact, we have
When \(|\frac{|y|}{{\textbf{r}}}-1|\ge \delta \), there hold
And for \( y\in \Omega _1^{+}\) and \(|\frac{|y|}{{\textbf{r}}}-1|\ge \delta \), we have
with \( \alpha =(\frac{N-2-m}{N-2},\frac{N-2}{2})\). Then we can get easily
If \(|\frac{|y|}{{\textbf{r}}}-1|\le \delta \), then
where \( \delta _1 \) is small constant. If \(|y- {\overline{x}}_1|\le \frac{\delta _1{\textbf{r}}}{k} \), it is easy to derive
for some small \( \delta _2 \). Therefore,
Hence
When \(|y- {\overline{x}}_1|\ge \frac{\delta _1{\textbf{r}}}{k} \), combing (A.22), we can get easily,
Thus we can get
Combining (A.17), (A.18), (A.20), (A.19), (A.21) and (A.23), we can get
\(\square \)
Combining Lemma A.1–A.3, we can get the following Proposition which gives the expression of \( I(W_{r,h,\Lambda }). \)
Proposition A.4
Suppose that K(|y|) satisfies \(({\textbf{H}}) \) and \(N\ge 5 \), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). Then we have
as \(k \rightarrow \infty \), where \( A_i,(i=1,2,3), B_4, B_5 \) are positive constants.
Proof
A direct result of Lemma A.1–A.3 is
with \(B_4=B_0B_1, B_5=B_0B_2\) are positive constants. From the expressions of \( \sigma _1(k), \sigma _2(k) \) and asymptotic expression of h, r as in (A.4), (1.10) , we can show that
can be absorbed in \( O\Big (\frac{1}{k^{\big (\frac{m(N-2)}{N-2-m}+\frac{2(N-3)}{N-1}+\sigma \big )}}\Big ). \)
Noting that \( m > \frac{N-2}{2} \) implies
thus provided with \( \epsilon _0, \sigma \) small enough, we can get
Since \( m\ge 2 \), we can check that
Thus we can get (A.24). \(\square \)
To get the expansions of \(\frac{F(r,h,\Lambda )}{\partial \Lambda }, \frac{F(r,h,\Lambda )}{\partial h}\), we need the following expansions for \( \frac{\partial I(W_{r,h,\Lambda })}{\partial \Lambda }, \frac{\partial I(W_{r,h,\Lambda })}{\partial h}\).
Proposition A.5
Suppose that K(|y|) satisfies \(({\textbf{H}}) \) and \(N\ge 5 \), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). We have
as \(k \rightarrow \infty \), where the constants \(B_i, i= 4,5 \) and \(A_i, i=2,3 \) are defined in Proposition A.4.
Proof
The proof of this proposition is standard and the reader can refer to [31] for details. \(\square \)
Proposition A.6
Suppose that K(|y|) satisfies \(({\textbf{H}}) \) and \(N\ge 5 \), \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k}\). Then we have
as \(k \rightarrow \infty \).
Proof
Recall
We know that
From (A.27), similar to the calculations in the proof of Proposition A.3, we can get
Then by some tedious but straightforward analysis, we can get
for some \(\epsilon _0\) small enough. In fact, we know that \( k \Big (\frac{k}{{\textbf{r}}}\Big )^{N-\epsilon _0}\) and \( h \frac{B_5 k}{r^{N-2} h^{N-3}({1-h^2})^{\frac{3}{2}}} \) can be absorbed in \(O\Big (\frac{1}{k^{\big (\frac{m(N-2)}{N-2-m}+\frac{(N-3)}{N-1}+\sigma \big )}}\Big )\) provided with m satisfying (1.6) and \( \epsilon _0, \sigma \) small enough. In fact, this is the reason why we need the assumption (1.6). Then we can get (A.25) directly. \(\square \)
Appendix B. Some basic estimates and lemmas
Lemma B.1
Under the condition \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k} \), for \( y\in \Omega _1^{+} \) there exists a constant C such that
with \( \alpha =(1, N-2) \).
Proof
For \( y\in \Omega _1^{+} \) and \( j= 2, \ldots , k \), we have
and
Then
\(\square \)
Lemma B.2
Under the condition \((r,h,\Lambda ) \in {{{\mathscr {S}}}_k} \), for \( y\in \Omega _1^{+} \) we have
with \( \alpha =(1, N-1) \).
Proof
The proof of Lemma B.2 is similar to Lemma B.1. We omit the details for concise. \(\square \)
For each fixed i and j, \(i\ne j \), we consider the following function
where \(\gamma _{1}\ge 1 \) and \(\gamma _{2}\ge 1 \) are two constants.
Lemma B.3
(Lemma B.1, [31]) For any constants \(0<\upsilon \le \min \{\gamma _{1},\gamma _{2}\} \), there is a constant \(C>0 \), such that
Lemma B.4
(Lemma B.2, [31]) For any constant \(0<\beta <N-2 \), there is a constant \(C>0 \), such that
Lemma B.5
Suppose that \(N\ge 5 \) and \(\tau \in (0, 2), y=(y_1, \ldots , y_N) \). Then there is a small \(\sigma >0 \), such that when \( y_3 \ge 0 \),
and when \( y_3 \le 0 \),
Proof
The proof of Lemma B.5 is similar to Lemma B.3 in [31]. Here we omit it. \(\square \)
Lemma B.6
Suppose that \(N\ge 5 \) and m satisfies (1.6). We have
provided with \(\sigma , \epsilon _1\) small enough.
Proof
It’s easy to show that
for \(m\ge 2\). In order to get (B.1), we just need to show
for some \(\sigma , \epsilon _1\) small. The problem to show (B.2) can be reduced to show that \(~ 6+ \frac{(N-3)}{N-1} < 3(\frac{N-2}{N-2-m} ) + 2\frac{N-2-m}{N-2}\), for m satisfying (1.6). This inequality follows by simple computations. This fact concludes the proof. \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Duan, L., Musso, M. & Wei, S. Doubling the equatorial for the prescribed scalar curvature problem on \({{\mathbb {S}}}^N\). Nonlinear Differ. Equ. Appl. 30, 40 (2023). https://doi.org/10.1007/s00030-023-00845-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-023-00845-z
Keywords
- Prescribed scalar curvature problem
- Finite dimensional Lyapunov–Schmidt reduction
- Lyapunov–Schmidt reduction
- Scalar curvature problem