Abstract
Let \((\mathcal {M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 8\). We are concerned with the following elliptic system
where \(\Delta _g=div_g \nabla \) is the Laplace–Beltrami operator on \(\mathcal {M}\), h(x) is a \(C^1\)-function on \(\mathcal {M}\), \(\varepsilon >0\) is a small parameter, \(\alpha ,\beta >0\) are real numbers, \((p,q)\in (1,+\infty )\times (1,+\infty )\) satisfies \(\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}\). Using the Lyapunov–Schmidt reduction method, we obtain the existence of multiple blowing-up solutions for the above problem.
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1 Introduction
Let \((\mathcal {M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 8\), where g denotes the metric tensor. We consider the following elliptic system
where \(\Delta _g=div_g \nabla \) is the Laplace–Beltrami operator on \(\mathcal {M}\), h(x) is a \(C^1\)-function on \(\mathcal {M}\), \(\varepsilon >0\) is a small parameter, \(\alpha ,\beta >0\) are real numbers, \((p,q)\in (1,+\infty )\times (1,+\infty )\) is a pair of numbers lying on the critical hyperbola
Without loss of generality, we assume that \(1<p\le \frac{N+2}{N-2}\le q\). Moreover, by (1.2), we have \(p>\frac{2}{N-2}\).
In the case \(u=v\), \(p=q\) and \(\alpha =\beta =1\), system (1.1) is reduced to the following equation
where \(N\ge 3\), \(2^*=\frac{2N}{N-2}\), \(\varepsilon \in \mathbb {R}\) is a small parameter. If \(h(x)=\frac{N-2}{4(N-1)}Scal_g\), where \(Scal_g\) is the scalar curvature of the manifold, equation (1.3) is intensively studied as the well-known Yamabe problem whose positive solutions u are such the scalar curvature of the conformal metric \(u^{2^*-2}g\) is constant, see e.g. [1, 31, 32, 34]. If \(h(x)\ne \frac{N-2}{4(N-1)}Scal_g\), Micheletti et al. [26] first proved that (1.3) has a single blowing-up solution, provided the graph of h(x) is distinct at some point from the graph of \(\frac{N-2}{4(N-1)}Scal_g\). Here, we say that a family of solutions \(u_\varepsilon \) of (1.3) blows up at a point \(\xi _0\in \mathcal {M}\) if there exists a family of points \(\xi _\varepsilon \in \mathcal {M}\) such that \(\xi _\varepsilon \rightarrow \xi _0\) and \(u_\varepsilon (\xi _\varepsilon )\rightarrow +\infty \) as \(\varepsilon \rightarrow 0\). Soon after, Deng [9] considered the existence of multiple blowing-up solutions which are separate from each other for (1.3). Chen [4] discovered the existence of clustered solutions which concentrate at one point in \(\mathcal {M}\) for (1.3). Moreover, Sign-changing bubble tower solutions of (1.3) have been established in [5, 27]. For more related results about (1.3), we refer the readers to [8, 10, 13, 30] and references therein.
Now, we return to the following elliptic system
called the Lane-Emden system, where \(N\ge 3\), (p, q) satisfies (1.2), \(\Omega \) is either a smooth bounded domain or \(\mathbb {R}^N\), and \(\mathcal {X}_{p,q}(\Omega )=\dot{W}^{2,\frac{p+1}{p}}(\Omega )\times \dot{W}^{2,\frac{q+1}{q}}(\Omega )\). System (1.4) has received remarkable attention for decades. When \(\Omega =\mathbb {R}^N\), by the Sobolev embedding theorem, there holds
with
Thus the following energy functional is well defined in \(\mathcal {X}_{p,q}(\mathbb {R}^N)\):
By applying the concentration compactness principle, Lions [25] found a positive least energy solution to (1.4) in \(\mathcal {X}_{p,q}(\mathbb {R}^N)\), which is radially symmetric and radially decreasing. Moreover, Wang [33] and Hulshof and Van der Vorst [19] independently proved that the uniqueness of the positive least energy solution \((U_{1,0}(x),V_{1,0}(z))\in \mathcal {X}_{p,q}(\mathbb {R}^N)\), and all the positive least energy solutions \(\big (U_{\delta ,\xi }(z),V_{\delta ,\xi }(x)\big )\) given by
Frank et al. [12] established the non-degeneracy of (1.4) at each least energy solution, that is, the linearized system around a least energy solution has precisely the \((N+1)\)-dimensional spaces of solutions in \(\mathcal {X}_{p,q}(\mathbb {R}^N)\). Furthermore, by using the Lyapunov–Schmidt reduction method and the non-degeneracy result obtained in [12], Guo et al. [18] established the existence and non-degeneracy of multiple blowing-up solutions to (1.4) with two potentials. For more investigations of system (1.4) with \(\Omega =\mathbb {R}^N\), we can see [7, 14].
If \(\Omega \) is a smooth bounded domain, much attention has been paid to study (1.4). Kim and Pistoia [22] first built multiple blowing-up solutions for the Lane-Emden system
where \(N\ge 8\), \(\varepsilon >0\), \(\alpha ,\beta _1,\beta _2\in \mathbb {R}\), \(1<p<\frac{N-1}{N-2}\), and (p, q) satisfies (1.2). Furthermore, using the local Pohozaev identities for the system, Guo et al. [16] proved the non-degeneracy of the blowing-up solutions to (1.5) constructed in [22]. Jin and Kim [20] studied the Coron’s problem for the critical Lane-Emden system, and established the existence, qualitative properties of positive solutions. More recently, inspired by [29], Guo and Peng [15] considered sign-changing solutions to the sightly supercritical Lane-Emden system with Neumann boundary conditions. For more classical results regarding Hamiltonian systems in bounded domains, the readers may refer to [3, 6, 17, 21, 28] for a good survey.
As far as we know, no existence result for the system (1.1)–(1.2) in the literature. Therefore, it is natural to ask that if the system possesses solutions on a smooth compact Riemannian manifold. Motivated by [22] and [26], in this paper, we give an affirmative answer for this question.
To state our main result, we first recall some definitions and results.
Definition 1.1
For \(k\ge 2\) to be a positive integer, let \((u_\varepsilon ,v_\varepsilon )\) be a family of solutions of (1.1)–(1.2), we say that \((u_\varepsilon ,v_\varepsilon )\) blows up and concentrates at point \(\bar{\xi ^0}=(\xi _1^0,\xi _2^0,\cdots , \xi _k^0)\in \mathcal {M}^k\) if there exist \(\bar{\xi }^\varepsilon =(\xi _1^\varepsilon ,\xi _2^\varepsilon ,\cdots , \xi _k^\varepsilon )\in \mathcal {M}^k\) and \((\delta _1^\varepsilon ,\delta _2^\varepsilon ,\cdots , \delta _k^\varepsilon )\in (\mathbb {R}^+)^k\) such that \(\xi _j^\varepsilon \rightarrow \xi _j^0\) and \(\delta _j^\varepsilon \rightarrow 0\) as \(\varepsilon \rightarrow 0\) for \(j=1,2,\cdots ,k\), and
where \(\Vert \cdot \Vert \) and \((W_{\delta ,\xi },H_{\delta ,\xi })\) are defined in (2.1) and (2.5).
Definition 1.2
[23, Definition 0.1] Let \(f\in C^1(\mathcal {M},\mathbb {R})\), for any given integer \(k\ge 2\), set \(\bar{\xi }=(\xi _1,\xi _2,\cdots , \xi _k)\), let \(\mathcal {C}_1,\mathcal {C}_2,\cdots , \mathcal {C}_k\subset \mathcal {M}\) be k mutually disjoint closed subsets of critical points of f, we say that \((\mathcal {C}_1,\mathcal {C}_2,\cdots , \mathcal {C}_k) \subset \mathcal {M}^k\) is a \(C^1\)-stable critical set of function \(F(\bar{\xi }):=\sum \limits _{j=1}^kf(\xi _j)\), if for any \(\varepsilon >0\), there exists \(\sigma >0\) such that if \(\Phi \in C^1(\mathcal {M}^k,\mathbb {R})\) with
then \(\Phi \) has at least one critical point \(\bar{\xi }\in \mathcal {M}^k\) with \(d_g(\xi _j,\mathcal {C}_j)<\varepsilon \), where \(d_g\) is the geodesic distance on \(\mathcal {M}\) with respect to the metric g.
Remark 1.3
[23, Remark 0.1] \((\mathcal {C}_1,\mathcal {C}_2,\cdots , \mathcal {C}_k)\subset \mathcal {M}^k\) is a \(C^1\)-stable critical set of function \(F(\bar{\xi })\) if one of the following conditions holds:
-
(i)
Every \(\mathcal {C}_j\) is a strict local minimum (or local maximum) set of f, \(j=1,2,\cdots ,k\).
-
(ii)
Every \(\mathcal {C}_j=\{\xi _j^0\}\) is an isolated critical point of \(f(\xi _j)\) with \(\deg (\nabla _g f,B_g(\xi _j^0,\rho ),0)\ne 0\) for some \(\rho >0\), where \(\deg \) is the Brouwer degree, and \(B_g(\xi _j^0,\rho )\) is the ball in \(\mathcal {M}\) centered at \(\xi _j^0\) with radius \(\rho \) with respect to the distance induced by the metric g, \(j=1,2,\cdots ,k\).
Let \(L_1,L_2,\cdots ,L_7\) be positive numbers defined by
Our main result states as follows.
Theorem 1.4
Let \((\mathcal {M},g)\) be a smooth compact Riemannian manifold, let h(x) be a \(C^1\)-function on \(\mathcal {M}\), (p, q) satisfies (1.2), for any given integer \(k\ge 2\), set \(\bar{\xi ^0}=(\xi _1^0,\xi _2^0,\cdots , \xi _k^0)\), let \(\xi _j^0\) be an isolated critical point of
with \(\varphi (\xi _j^0)>0\) and \(\deg (\nabla _g \varphi ,B_g(\xi _j^0,\rho ),0)\ne 0\) for some \(\rho >0\), \(j=1,2,\cdots ,k\), Assume that one of the following conditions holds:
-
(i)
\(\frac{N}{N-2}<p<\frac{N+2}{N-2}\) and \(N\ge 8\);
-
(ii)
\(p=\frac{N+2}{N-2}\) and \(N\ge 10\);
-
(iii)
\(1<p<\frac{N}{N-2}\) and \(N\ge 8\).
Then for \(\varepsilon >0\) small enough, system (1.1) admits a family of solutions \((u_\varepsilon ,v_\varepsilon )\), which blows up and concentrates at \(\bar{\xi }^0\) as \(\varepsilon \rightarrow 0\).
Remark 1.5
Under the assumptions on p, q and N of Theorem 1.4, we have that \(L_i<+\infty \) for \(i=1,2,\cdots ,7\).
Remark 1.6
From the proof of Theorem 1.4 (see Sect. 3), it’s easy to find that if
then Theorem 1.4 still holds true. However, in the proof of Proposition 3.1, we have to impose \(\alpha ,\beta >0\) to guarantee the continuous embedding, see e.g. (4.9)–(4.10) and (5.18)–(5.19).
Remark 1.7
If \(u=v\), \(p=q=\frac{N+2}{N-2}\), \(\alpha =\beta =1\), then
and Theorem 1.4 is exactly the conclusion obtained in [9, Theorem 1.1].
The proof of our result relies on a well known finite dimensional Lyapunov–Schmidt reduction method, introduced in [2, 11]. The paper is organized as follows. In Sect. 2, we introduce the framework and present some preliminary results. The proof of Theorem 1.4 is given in Sect. 3. In Sect. 4, we perform the finite dimensional reduction, and Sect. 5 is devoted to the reduced problem. Throughout the paper, \(C,C_i\), \(i\in \mathbb {N}^+\) denote positive constants possibly different from line to line.
2 The Framework and Preliminary Results
Concerning the least energy solution \((U_{1,0}(z),V_{1,0}(z))\) of (1.4) with \(\Omega =\mathbb {R}^N\), we have the following asymptotic behaviour and non-degeneracy result.
Lemma 2.1
[19, Theorem 2] Assume that \(1<p\le \frac{N+2}{N-2}\). If \(r\rightarrow +\infty \), there hold
and
Lemma 2.2
[21, Lemma 2.2] Assume that \(1<p\le \frac{N+2}{N-2}\). If \(r\rightarrow +\infty \), there hold
and
Lemma 2.3
[15, Remark 2.3] Assume that \(1<p\le \frac{N+2}{N-2}\). If \(r\rightarrow +\infty \), there hold
and
Lemma 2.4
[12, Theorem 1] Set
and
Then the space of solutions to the linear system
is spanned by
Moreover, we have the following elementary inequality.
Lemma 2.5
[24, Lemma 2.1] For any \(a>0\), b real, there holds
Now, we recall some definitions and results about the compact Riemannian manifold \((\mathcal {M},g)\).
Definition 2.6
Let \((\mathcal {M},g)\) be a smooth compact Riemannian manifold. On the tangent bundle of \(\mathcal {M}\), define the exponential map \(\exp : T \mathcal {M}\rightarrow \mathcal {M}\), which has the following properties:
-
(i)
\(\exp \) is of class \(C^\infty \);
-
(ii)
there exists a constant \(r_0>0\) such that \(\exp _\xi |_{B(0,r_0)}\rightarrow B_g(\xi ,r_0)\) is a diffeomorphism for all \(\xi \in \mathcal {M}\).
Fix such \(r_0\) in this paper with \(r_0<i_g/{2}\), where \(i_g\) denotes the injectivity radius of \((\mathcal {M},g)\). For any \(1<s<+\infty \) and \(u\in L^s(\mathcal {M})\), we denote the \(L^s\)-norm of u by
where \(d v_g=\sqrt{\det g}{\text {d}}z\) is the volume element on \(\mathcal {M}\) associated to the metric g. We introduce the Banach space
equipped with the norm
Denote by \(\mathcal {I}^*\) the formal adjoint operator of the embedding \(\mathcal {I}:\mathcal {X}_{q,p}(\mathcal {M})\hookrightarrow L^{p+1}(\mathcal {M})\times L^{q+1}(\mathcal {M})\). By the Calderón-Zygmund estimate, the operator \(\mathcal {I}^*\) maps \(L^{\frac{p+1}{p}}(\mathcal {M})\times L^{\frac{q+1}{q}}(\mathcal {M})\) to \(\mathcal {X}_{p,q}(\mathcal {M})\). Then we rewrite problem (1.1) as
where \(f_\varepsilon (u):=u_+^{p-\alpha \varepsilon }\), \(g_\varepsilon (u):=u_+^{q-\beta \varepsilon }\) and \(u_+=\max \{u,0\}\). Moreover, by the Sobolev embedding theorem, we have
and
Let \(\chi \) be a smooth cutoff function such that \(0\le \chi \le 1\) in \(\mathbb {R}^+\), \(\chi =1\) in \([0,r_0/{2}]\), and \(\chi =0\) out of \([r_0,+\infty ]\). For any \(\xi \in \mathcal {M}\) and \(\delta >0\), we define the following functions on \(\mathcal {M}\)
and
for \(i=0,1,\cdots , N\), where \((\Psi _{1,0}^i,\Phi _{1,0}^i)\) is given in Lemma 2.4.
For any \(\varepsilon >0\) and \(\bar{t}=(t_1,t_2,\cdots ,t_k)\in (\mathbb {R}^+)^k\), we set
for fixed small \(\varrho _1>0\). Moreover, for \(\varrho _2\in (0,1)\), we define the configuration space \(\Lambda \) by
Let \(\mathcal {Y}_{\bar{\delta },\bar{\xi }}\) and \(\mathcal {Z}_{\bar{\delta },\bar{\xi }}\) be two subspaces of \(\mathcal {X}_{p,q}(\mathcal {M})\) given as
and
where
for any \((u,v),(\varphi ,\psi )\in \mathcal {X}_{p,q}(\mathcal {M})\).
Lemma 2.7
There exists \(\varepsilon _0>0\) such that for any \(\varepsilon \in (0,\varepsilon _0)\), \(\mathcal {X}_{p,q}(\mathcal {M})=\mathcal {Y}_{\bar{\delta },\bar{\xi }}\oplus \mathcal {Z}_{\bar{\delta },\bar{\xi }}\).
Proof
We shall prove that for any \((\Psi ,\Phi )\in \mathcal {X}_{p,q}(\mathcal {M})\), there exists unique pair \((\Psi _0,\Phi _0)\in \mathcal {Z}_{\bar{\delta },\bar{\xi }}\) and coefficients \(c_{10},c_{11},\cdots ,c_{1N}, c_{20},c_{21},\cdots ,c_{2N},\cdots ,c_{k0},c_{k1},\cdots ,c_{kN}\) such that
The requirement that \((\Psi _0,\Phi _0)\in \mathcal {Z}_{\bar{\delta },\bar{\xi }}\) is equivalent to demanding
for any \(i=0,1,\cdots ,N\) and \(j=1,2,\cdots ,k\).
We estimate the integral on the right-hand side of (2.8). By standard properties of the exponential map, there exists \(C>0\) such that for any \(\xi \in \mathcal {M}\), \(\delta >0\), \(z\in B(0,r_0/\delta )\), and \(i,j,k\in \mathbb {N}^+\), there hold
where \(g_{\delta ,\xi }(z)=\exp _\xi ^*g(\delta z)\) and \((\Gamma _{\delta ,\xi })_{ij}^k\) stand for the Christoffel symbols of the metric \(g_{\delta ,\xi }\). Taking into account that there holds
by Lemma 2.4 and \(dg(\xi _j,\xi _m)>2r_0\) for \(j\ne m\), we have
and
where \(\chi _{\delta _j}(x)=\chi ({\delta _j|z|})\) and \(h_{\delta _j,\xi _j}(z)=h(\exp _{\xi _j}(\delta _j z))\). Similarly, we have
and
By plugging (2.11)–(2.14) into (2.8), we can see that the coefficients \(c_{lm}\) are uniquely determined for \(l=0,1,\cdots ,N\) and \(m=1,2,\cdots ,k\). By virtue of (2.7), so is \((\Psi _0,\Phi _0)\).
On the other hand, \(\mathcal {Y}_{\bar{\delta },\bar{\xi }}\) and \(\mathcal {Z}_{\bar{\delta },\bar{\xi }}\) are clearly closed subspaces of \(\mathcal {X}_{p,q}(\mathcal {M})\), Therefore, they are topological complements of each other. \(\square \)
3 Scheme of the Proof of Theorem 1.4
We look for solutions of system (1.1), or equivalently of (2.2), of the form
where \(\bar{\delta }\) is as in (2.6), \((W_{\delta _j,\xi _j},H_{\delta _j,\xi _j})\) is as in (2.5), and \((\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\Phi _{\varepsilon ,\bar{t},\bar{\xi }})\in \mathcal {Z}_{\bar{\delta },\bar{\xi }}\). By Lemma 2.7, \(\mathcal {X}_{p,q}(\mathcal {M})=\mathcal {Y}_{\bar{\delta },\bar{\xi }}\oplus \mathcal {Z}_{\bar{\delta },\bar{\xi }}\). Then we define the projections \(\Pi _{\bar{\delta },\bar{\xi }}\) and \(\Pi _{\bar{\delta },\bar{\xi }}^\bot \) of the Sobolev space \(\mathcal {X}_{p,q}(\mathcal {M})\) onto \(\mathcal {Y}_{\bar{\delta },\bar{\xi }}\) and \(\mathcal {Z}_{\bar{\delta },\bar{\xi }}\) respectively. Therefore, we have to solve the couples of equations
and
The first step in the proof consists in solving equation (3.3). This requires Proposition 3.1 below, whose proof is postponed to Sect. 4.
Proposition 3.1
Under the assumptions of Theorem 1.4, if \((\bar{\delta },\bar{\xi })\in \Lambda \) and \(\bar{\delta }\) is as in (2.6), then for any \(\varepsilon >0\) small enough, equation (3.3) admits a unique solution \((\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\Phi _{\varepsilon ,\bar{t},\bar{\xi }})\) in \(\mathcal {Z}_{\bar{\delta },\bar{\xi }}\), which is continuously differentiable with respect to \(\bar{t}\) and \(\bar{\xi }\), such that
We now introduce the energy functional \(\mathcal {J}_\varepsilon \) defined on \(\mathcal {X}_{p,q}(\mathcal {M})\) by
It is clear that the critical points of \(\mathcal {J}_\varepsilon \) are the solutions of system (1.1). Moreover,
for any \((u,v),(\varphi ,\psi )\in \mathcal {X}_{p,q}(\mathcal {M})\). We also define the functional \(\widetilde{\mathcal {J}}_\varepsilon :(\mathbb {R}^+)^k\times \mathcal {M}^k\rightarrow \mathbb {R}\)
where \((\mathcal {W}_{\bar{\delta },\bar{\xi }},\mathcal {H}_{\bar{\delta },\bar{\xi }})\) is as (3.1), \((\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\Phi _{\varepsilon ,\bar{t},\bar{\xi }})\) is given in Proposition 3.1.
Definition 3.2
For a given \(C^1\)-function \(\varphi _\varepsilon \), we say that the estimate \(\varphi _\varepsilon =o(\varepsilon )\) is \(C^1\)-uniform if there hold \(\varphi _\varepsilon =o(\varepsilon )\) and \(\nabla \varphi _\varepsilon =o(\varepsilon )\) as \(\varepsilon \rightarrow 0\).
We solve equation (3.2) in Proposition 3.3 below whose proof is postponed to Sect. 5.
Proposition 3.3
(i) Under the assumptions of Theorem 1.4, if \(\bar{\delta }\) is as in (2.6), for any \(\varepsilon >0\) small enough, if \((\bar{t},\bar{\xi })\) is a critical point of the functional \(\widetilde{\mathcal {J}}_\varepsilon \), then \(\big (\mathcal {W}_{\bar{\delta },\bar{\xi }}+\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\mathcal {H}_{\bar{\delta },\bar{\xi }}+\Phi _{\varepsilon ,\bar{t},\bar{\xi }}\big )\) is a solution of system (1.1), or equivalently of (2.2).
(ii) Under the assumptions of Theorem 1.4, there holds
as \(\varepsilon \rightarrow 0\), \(C^1\)-uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\), where
and
with \(L_1,L_3,L_6,L_7\) are positive constants given in (1.6), \(\varphi (\xi _j)\) is defined as (1.7), \(j=1,2,\cdots ,k\).
We now prove Theorem 1.4 by using Propositions 3.1 and 3.3.
Proof of Theorem 1.4
Define \(\widetilde{\mathcal {J}}:(\mathbb {R}^+)^k\times \mathcal {M}^k\rightarrow \mathbb {R}\) by
where \(\widetilde{C}=\big (\frac{\alpha }{(p+1)^2} +\frac{\beta }{(q+1)^2}\big )\frac{NL_1}{2}\) and \(L_1,L_3>0\) are given in (1.6). Since \(\xi _j^0\) is an isolated critical point of the function \(\varphi (\xi _j)\) with \(\varphi (\xi _j^0)>0\), and set \(t_j^0=\frac{\widetilde{C}}{L_3\varphi (\xi _j^0)}\), then \(t_j^0>0\) and \((t_j^0,\xi _j^0)\) is an isolated critical point of \(f(t_j,\xi _j)\). Moreover, by \(\deg (\nabla _g \varphi ,B_g(\xi _j^0,\rho ),0)\ne 0\) for some \(\rho >0\), we obtain \(\deg (\nabla _g f,B_g(\xi _j^0,\rho ),0)\ne 0\), \(j=1,2,\cdots ,k\). Hence, by Remark 1.3, \((\bar{t^0},\bar{\xi ^0})\) is a \(C^1\)-stable critical set of \(\widetilde{\mathcal {J}}\), where \(\bar{t^0}=(t_1^0,t_2^0,\cdots ,t_k^0)\) and \(\bar{\xi ^0}=(\xi _1^0,\xi _2^0,\cdots ,\xi _k^0)\). Using Proposition 3.3, we have
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\). By standard properties of the Brouwer degree, it follows that there exists a family of critical points \((\bar{t^\varepsilon },\bar{\xi ^\varepsilon })\) of \(\widetilde{\mathcal {J}}_\varepsilon \) converging to \((\bar{t^0},\bar{\xi ^0})\) as \(\varepsilon \rightarrow 0\). Using Proposition 3.3 again, we can see that the function \((u_\varepsilon ,v_\varepsilon )=\big (\mathcal {W}_{\bar{\delta ^\varepsilon },\bar{\xi ^\varepsilon }}+\Psi _{\varepsilon ,\bar{t^\varepsilon },\bar{\xi ^\varepsilon }},\mathcal {H}_{\bar{\delta ^\varepsilon },\bar{\xi ^\varepsilon }}+\Phi _{\varepsilon ,\bar{t^\varepsilon },\bar{\xi ^\varepsilon }}\big )\) is a pair of solutions of system (1.1) for any \(\varepsilon >0\) small enough, where \(\bar{\delta ^\varepsilon }\) is as in (2.6). Moreover, \((u_\varepsilon ,v_\varepsilon )\) blows up and concentrates at \(\bar{\xi ^0}\) at \(\varepsilon \rightarrow 0\). This ends the proof.
\(\square \)
4 Proof of Proposition 3.1
This section is devoted to the proof of Proposition 3.1. For any \(\varepsilon >0\), \(\bar{t}\in (\mathbb {R}^+)^k\), and \(\bar{\xi }\in \mathcal {M}^k\), if \(\bar{\delta }\) is as in (2.6), we introduce the map \(\mathcal {L}_{\varepsilon ,\bar{t},\bar{\xi }}:\mathcal {Z}_{\bar{\delta },\bar{\xi }}\rightarrow \mathcal {Z}_{\bar{\delta },\bar{\xi }}\) defined by
It’s easy to check that \(\mathcal {L}_{\varepsilon ,\bar{t},\bar{\xi }}\) is well defined in \(\mathcal {Z}_{\bar{\delta },\bar{\xi }}\). Next, we prove the invertibility of this map.
Lemma 4.1
Under the assumptions on p, q and N of Theorem 1.4, if \((\bar{\delta },\bar{\xi })\in \Lambda \) and \(\bar{\delta }\) is as in (2.6), then for any \(\varepsilon >0\) small enough, and \((\Psi ,\Phi )\in \mathcal {Z}_{\bar{\delta },\bar{\xi }}\), there holds
where \(\mathcal {L}_{\varepsilon ,\bar{t},\xi }(\Psi ,\Phi )\) is as in (4.1).
Proof
We assume by contradiction that there exist a sequence \(\varepsilon _n\rightarrow 0\) as \(n\rightarrow +\infty \), \((\bar{\delta _n},\bar{\xi _n})\in \Lambda \), \(\bar{t_n}=(t_{1n},t_{2n},\cdots ,t_{kn})\in (\mathbb {R}^+)^k\), \(\bar{\xi _n}=(\xi _{1n},\xi _{2n},\cdots ,\xi _{kn})\in \mathcal {M}^k\), and a sequence of functions \((\Psi _n,\Phi _n)\in \mathcal {Z}_{\bar{\delta _n},\bar{\xi _n}}\) such that
Then \(\Vert \Psi _n\Vert _{{q+1}}\le C\) and \(\Vert \Phi _n\Vert _{{p+1}}\le C\).
Step 1: For any \(n\in \mathbb {N}^+\) and \(j=1,2,\cdots ,k\), let
where \(\chi \) is a cutoff function as in (2.5). A direct computations shows
and
Hence, \((\widetilde{\Psi }_n,\widetilde{\Phi }_n)\) is bounded in \( \mathcal {X}_{p,q}(\mathbb {R}^N)\). Up to a subsequence, there exists \((\widetilde{\Psi },\widetilde{\Phi })\in \mathcal {X}_{p,q}(\mathbb {R}^N)\) such that \((\widetilde{\Psi }_n,\widetilde{\Phi }_n)\rightharpoonup (\widetilde{\Psi },\widetilde{\Phi })\) in \(\mathcal {X}_{p,q}(\mathbb {R}^N)\), \((\widetilde{\Psi }_n,\widetilde{\Phi }_n)\rightarrow (\widetilde{\Psi },\widetilde{\Phi })\) in \(L_{loc}^{s}(\mathbb {R}^N)\times L_{loc}^{t}(\mathbb {R}^N)\) for any \((s,t)\in [1,q+1]\times [1,p+1]\), and \((\widetilde{\Psi }_n,\widetilde{\Phi }_n)\rightarrow (\widetilde{\Psi },\widetilde{\Phi })\) almost everywhere in \(\mathbb {R}^N\). For convenience, we denote \((P_n,K_n)=\mathcal {L}_{\varepsilon _n,\bar{t_n},\bar{\xi _n}}(\Psi _n,\Phi _n)\). Furthermore, by \((P_n,K_n) \in \mathcal {Z}_{\bar{\delta _{n}},\bar{\xi _n}}\), there exist \(c_{1n}^0,c_{1n}^1,\cdots ,c_{1n}^N\), \(c_{2n}^0,c_{2n}^1,\cdots ,c_{2n}^N\), \(\cdots \), \(c_{kn}^0,c_{kn}^1,\cdots ,c_{kn}^N\) such that
which also reads
Using \((\Psi _n,\Phi _n) \in \mathcal {Z}_{\bar{\delta _{n}},\bar{\xi _n}}\) again, by an easy change of variable, for \(i=0,1,\cdots ,N\) and \(j=1,2\cdots ,k\), we have
where \(g_{n}(z)=\exp _{\xi _{jn}}^*g(\delta _{jn}z)\), \(\chi _{n}(z)=\chi ({\delta _{jn}|z|})\) and \(h_{n}(z)=h(\exp _{\xi _{jn}}(\delta _{jn}z))\). By Lemma 2.4, passing to the limit for the above equality, we obtain
Step 2: For any \(l=0,1,\cdots ,N\) and \(m=1,2,\cdots ,k\), \(c_{mn}^l\rightarrow 0\) as \(n\rightarrow \infty \). For any \(n\in \mathbb {N}^+\), since \((\Psi _n,\Phi _n)\) and \( (P_n,K_n)\) belong to \( \mathcal {Z}_{\bar{\delta _{n}},\bar{\xi _n}}\), multiplying (4.2) by \((\Psi _{\delta _{jn},\xi _{jn}}^i,\Phi _{\delta _{jn},\xi _{jn}}^i)\), \(0\le i\le N\), \(1\le j\le k\), using (2.11)–(2.14), we have
Moreover, by (4.4), we have
It follows from (4.5) and (4.6) that for any \(l=0,1,\cdots ,N\) and \(m=1,2,\cdots ,k\), \(c_{mn}^l\rightarrow 0\) as \(n\rightarrow \infty \).
Step 3: \((\widetilde{\Psi },\widetilde{\Phi })=(0,0)\). For any \(j=1,2,\cdots ,k\), there hold
and
Thus we obtain a system of equations satisfied by \((\widetilde{\Psi }_n,\widetilde{\Phi }_n)\). For any \((\varphi ,\psi )\in C_0^\infty (\mathbb {R}^N)\times C_0^\infty (\mathbb {R}^N)\) and \(j=1,2,\cdots ,k\), by the dominated convergence theorem, we obtain
and
Using (4.3), \(\Vert (P_n,K_n)\Vert \rightarrow 0\), \(c_{mn}^l\rightarrow 0\) as \(n\rightarrow \infty \) for any \(l=0,1,\cdots ,N\) and \(m=1,2,\cdots ,k\), we deduce that \((\widetilde{\Psi },\widetilde{\Phi }) \) satisfies
This together with (4.4) and Lemma 2.4 yields that \((\widetilde{\Psi },\widetilde{\Phi })=(0,0)\).
Step 4: \(\Vert \mathcal {I}^*\big (f'_{\varepsilon _n}(\mathcal {H}_{\bar{\delta _n},\bar{\xi _n}})\Phi _n,g'_{\varepsilon _n}(\mathcal {W}_{\bar{\delta _n},\bar{\xi _n}})\Psi _n\big )\Vert \rightarrow 0\) as \(n\rightarrow \infty \). By (2.3), we know
For any fixed \(R>0\) and \(j=1,2,\cdots ,k\), by the Hölder inequality, \(\widetilde{\Phi }_n\rightarrow 0\) in \(L_{loc}^{\frac{p+1}{1+\alpha {\varepsilon _n}}}({\mathbb {R}^N})\) and \(\widetilde{\Psi }_n\rightarrow 0\) in \(L_{loc}^{\frac{q+1}{1+\beta {\varepsilon _n}}}({\mathbb {R}^N})\), we have
and
From the above arguments, we get \(\Vert (\Psi _n,\Phi _n)\Vert \rightarrow 0\) as \(n\rightarrow +\infty \), which is an absurd. Thus, we complete the proof. \(\square \)
For any \(\varepsilon >0\) small enough, \(\bar{t}\in (\mathbb {R}^+)^k\), and \(\bar{\xi }\in \mathcal {M}^k\), if \(\bar{\delta }\) is as in (2.6), then equation (3.3) is equivalent to
where
and
In the following lemma, we estimate the reminder term \(\mathcal {R}_{\varepsilon ,\bar{t},\bar{\xi }}\).
Lemma 4.2
Under the assumptions on p, q and N of Theorem 1.4, if \((\bar{\delta },\bar{\xi })\in \Lambda \) and \(\bar{\delta }\) is as in (2.6), then for any \(\varepsilon >0\) small enough, there holds
where \(\mathcal {R}_{\varepsilon ,\bar{t},\bar{\xi }}\) is as in (4.8).
Proof
By (2.3), we know there exists \(C>0\) such that for \(\varepsilon >0\) small enough, \(\bar{t}\in (\mathbb {R}^+)^k\), and \(\bar{\xi }\in \mathcal {M}^k\), there holds
By an easy change of variable, and using Lemma 2.4, for any \(j=1,2,\cdots ,k\), we have
where \(g_{\delta _j,\xi _j}(z)=\exp _{\xi _j}^*g(\delta _jz)\), \(\chi _{\delta _j}(z)=\chi ({\delta _j|z|})\) and \(h_{\delta _j,\xi _j}(z)=h(\exp _{\xi _j}(\delta _jz))\). We are led to estimate each \(A_i\), \(i=1,2,\cdots ,6\). First, for any fixed \(R>0\) large enough and \(j=1,2,\cdots ,k\), by Lemma 2.1 and Taylor formula, we have
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\xi _j\in \mathcal {M}\) and \(t_j\in [a,b]\), \(0<a<b<+\infty \), where we have used the fact that \(N<\frac{(N-2)(p+1)^2}{p}\), since \(p>\frac{2}{N-2}\). Using Lemma 2.1 and Taylor formula again, for \(j=1,2,\cdots ,k\), we obtain
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\xi _j\in \mathcal {M}\) and \(t_j\in [a,b]\). Since \(N\ge 8\), then \(A_2\le |\varepsilon \log \varepsilon |^{\frac{p+1}{p}}\). For any fixed \(R>0\) large enough and \(j=1,2,\cdots ,k\), it follows from (2.9) and (2.10) that
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\xi _j\in \mathcal {M}\) and \(t_j\in [a,b]\), where we have used the fact that \(N\ge 8\) and \(p>1\). Since there hold \(|\chi '_{\delta _j}|\le C\delta _j\) and \(|\chi ''_{\delta _j}|\le C\delta ^2_j\) for any \(j=1,2,\cdots ,k\), we have
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\xi _j\in \mathcal {M}\) and \(t_j\in [a,b]\). Moreover, for any fixed \(R>0\) large enough and \(j=1,2,\cdots ,k\), it’s easy to obtain
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\xi _j\in \mathcal {M}\) and \(t_j\in [a,b]\). From the above arguments, we obtain \(I_j=O(\varepsilon |\log \varepsilon |)\) for any \(j=1,2,\cdots ,k\).
Similarly, we can prove that
For any fixed \(R>0\) large enough and \(j=1,2,\cdots ,k\), by \(N\ge 8\) and \(q>1\), we have
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\xi _j\in \mathcal {M}\) and \(t_j\in [a,b]\). Similar arguments as above, we have
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\xi _j\in \mathcal {M}\) and \(t_j\in [a,b]\). Hence \(II_j=O(\varepsilon |\log \varepsilon |)\) for any \(j=1,2,\cdots ,k\). This ends the proof. \(\square \)
We now prove Proposition 3.1 by using Lemmas 4.1 and 4.2.
Proof of Proposition 3.1
For any \(\varepsilon >0\) small enough, \(\bar{t}\in (\mathbb {R}^+)^k\), and \(\bar{\xi }\in \mathcal {M}^k\), if \(\bar{\delta }\) is as in (2.6), we define the map \(\mathcal {T}_{\varepsilon ,\bar{t},\bar{\xi }}:\mathcal {Z}_{\bar{\delta },\bar{\xi }}\rightarrow \mathcal {Z}_{\bar{\delta },\bar{\xi }}\) by
where \(\mathcal {L}_{\varepsilon ,\bar{t},\bar{\xi }}\), \(\mathcal {N}_{\varepsilon ,\bar{t},\bar{\xi }}\) and \(\mathcal {R}_{\varepsilon ,\bar{t},\bar{\xi }}\) are as in (4.1), (4.7) and (4.8), respectively. We also set
where \(\gamma >0\) is a fixed constant large enough. We prove that the map \(\mathcal {T}_{\varepsilon ,\bar{t},\bar{\xi }}\) admits a fixed point \((\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\Phi _{\varepsilon ,\bar{t},\bar{\xi }})\). Therefore, we shall prove that, for any \(\varepsilon >0\) small, there hold:
(i) \(\mathcal {T}_{\varepsilon ,\bar{t},\bar{\xi }}(\mathcal {B}_{\varepsilon ,\bar{t},\bar{\xi }}(\gamma ))\subset \mathcal {B}_{\varepsilon ,\bar{t},\bar{\xi }}(\gamma )\);
(ii) \(\mathcal {T}_{\varepsilon ,\bar{t},\bar{\xi }}\) is a contraction map on \(\mathcal {B}_{\varepsilon ,\bar{t},\bar{\xi }}(\gamma )\).
For (i), by (2.3) and Lemma 4.1, for any \(\varepsilon >0\) small enough, and \((\Psi ,\Phi )\in \mathcal {B}_{\varepsilon ,\bar{t},\bar{\xi }}(\gamma )\), we have
By the mean value formula, Lemmas 2.5, 4.2, and the Sobolev embedding theorem, we obtain
and
where we have used the fact that \(\Vert W_{\delta _j,\xi _j}\Vert _{q+1}<+\infty \) for any \(1<p\le \frac{N+2}{N-2}\le q\) and \(j=1,2,\cdots ,k\). So we have (i).
Similarly, by (2.3) and Lemma 4.1, for any \(\varepsilon >0\) small enough, and \((\Psi _1,\Phi _1), (\Psi _2,\Phi _2)\in \mathcal {B}_{\varepsilon ,\bar{t},\bar{\xi }}(\gamma )\), we have
By the mean value formula, Lemma 2.5, and the Sobolev embedding theorem, we obtain
and
By Lemma 4.2, we know \(C \gamma \Vert \mathcal {R}_{\varepsilon ,\bar{t},\bar{\xi }}\Vert ,C \gamma ^{p-1-\alpha \varepsilon }\Vert \mathcal {R}_{\varepsilon ,\bar{t},\bar{\xi }}\Vert ^{p-1-\alpha \varepsilon },C \gamma ^{q-1-\beta \varepsilon }\Vert \mathcal {R}_{\varepsilon ,\bar{t},\bar{\xi }}\Vert ^{q-1-\beta \varepsilon }\in (0,1)\). This proves (ii). Finally, by using the implicit function theorem, we can prove the regularity of \((\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\Phi _{\varepsilon ,\bar{t},\bar{\xi }})\) with respect to \(\bar{t}\) and \(\bar{\xi }\). Thus we complete the proof. \(\square \)
5 Proof of Proposition 3.3
This section is devoted to the proof of Proposition 3.3. As a first step, we have
Lemma 5.1
Under the assumptions on p, q and N of Theorem 1.4, if \(\bar{\delta }\) is as in (2.6), then for any \(\varepsilon >0\) small enough, if \((\bar{t},\bar{\xi })\) is a critical point of the functional \(\widetilde{\mathcal {J}}_\varepsilon \), then \(\big (\mathcal {W}_{\bar{\delta },\bar{\xi }}+\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\mathcal {H}_{\bar{\delta },\bar{\xi }}+\Phi _{\varepsilon ,\bar{t},\bar{\xi }}\big )\) is a solution of system (1.1), or equivalently of (2.2).
Proof
Let \((\bar{t},\bar{\xi })\) is a critical point of \(\widetilde{\mathcal {J}}_\varepsilon \), where \(\bar{t}=(t_1,t_2,\cdots ,t_k)\in (\mathbb {R}^+)^k\) and \(\bar{\xi }=(\xi _1,\xi _2,\cdots ,\xi _k)\in \mathcal {M}^k\). Let \(\bar{\xi }(y)=\big (\exp _{\xi _1}(y^1),\exp _{\xi _2}(y^2),\cdots ,\exp _{\xi _k}(y^k)\big )\), \(y=(y^1,y^2,\cdots ,y^k)\in B(0,r)^k\), and \(\xi _j(y^j)=\exp _{\xi _j}(y^j)\) for any \(j=1,2,\cdots ,k\), then \(\bar{\xi }(0)=\bar{\xi }\). Since \((\bar{t},\bar{\xi })\) is a critical point of \(\widetilde{\mathcal {J}}_\varepsilon \), for any \(m=1,2,\cdots ,k\) and \(l=1,2,\cdots ,N\), there hold
and
For any \((\varphi ,\psi )\in \mathcal {X}_{p,q}(\mathcal {M})\), by Proposition 3.1, there exist some constants \(c_{10},c_{11},\cdots ,c_{1N}\), \(c_{20},c_{21},\cdots ,c_{2N}\), \(\cdots \), \(c_{k0},c_{k1},\cdots ,c_{kN}\) such that
Let \(\partial _s\) denote \(\partial _ {t_m}\) or \(\partial _{y^m_l}\) for any \(m=1,2,\cdots ,k\) and \(l=1,2,\cdots ,N\). Then
We prove that if we compute (5.1) at \(y=0\), then for any \(\varepsilon >0\) small enough, there holds
Since \((\bar{t},\bar{\xi })\) is a critical point of \(\widetilde{\mathcal {J}}_\varepsilon \), then
For any \(m=1,2,\cdots ,k\) and \(l=1,2,\cdots ,N\), we can easily check that there hold
and
where \(\Vert (R_1,R_2)\Vert =o(\varepsilon ^{\frac{\vartheta }{2}})\) as \(\varepsilon \rightarrow 0\) for all \(\vartheta \in (0,1)\). Using (2.11)–(2.14), we have
and
where \(\chi _{\delta _m}(x)=\chi (\delta _m|x|)\). For any \(\vartheta \in (0,1)\), with the aid of Proposition 3.1, it’s easy to check
and
Therefore, by (5.2) and (5.5)–(5.9), we deduce that the linear system in (5.1) has only a trivial solution when \(y=0\) provided that \(\varepsilon >0\) small enough. This ends the proof. \(\square \)
In the next lemma, we give the asymptotic expansion of \( \mathcal {J}_{\varepsilon }(\mathcal {W}_{\bar{\delta },\bar{\xi }},\mathcal {H}_{\bar{\delta },\bar{\xi }})\) as \(\varepsilon \rightarrow 0\) for \((\bar{\delta },\bar{\xi })\in \Lambda \), where \(\bar{\delta }\) is as in (2.6).
Lemma 5.2
Under the assumptions on p, q and N of Theorem 1.4, if \((\bar{\delta },\bar{\xi })\in \Lambda \) and \(\bar{\delta }\) is as in (2.6), then there holds
as \(\varepsilon \rightarrow 0\), \(C^1\)-uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\), where the function \(\Psi _k(\bar{t},\bar{\xi })\) is defined as (3.5), \(c_1\) and \(c_2\) are given in (3.6).
Proof
For any \(\xi \in \mathcal {M}\), there holds
as \(r\rightarrow 0\), where \(\omega _{N-1}\) is the volume of the unit sphere in \(\mathbb {R}^N\). Furthermore, by standard properties of the exponential map, the reminder \(O(r^4)\) can be made \(C^1\)-uniform with respect to \(\xi \). Under the assumptions on p, q and N of Theorem 1.4, we can compute
and
as \(\varepsilon \rightarrow 0\), \(C^1\)-uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\), where \(g_{\delta _j,\xi _j}(z)=\exp _{\xi _j}^*g(\delta _jz)\), \(\chi _{\delta _j}(z)=\chi ({\delta _j|z|})\), and \(h_{\delta _j,\xi _j}(z)=h(\exp _{\xi _j}(\delta _jz))\). Using the Taylor formula, we have
and
as \(\varepsilon \rightarrow 0\), \(C^1\)-uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\). Similarly, we can prove that
and
as \(\varepsilon \rightarrow 0\), \(C^1\)-uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\), where we have used the fact that \(N\ge 10\) if \(p=\frac{N}{N-2}\) and \(N\ge 12\) if \(p<\frac{N}{N-2}\). From (5.10)–(5.17), we conclude the result. \(\square \)
We now give the asymptotic expansion of the function \(\widetilde{\mathcal {J}}_\varepsilon \) defined in (3.4) as \(\varepsilon \rightarrow 0\).
Lemma 5.3
Under the assumptions on p, q and N of Theorem 1.4, if \((\bar{\delta },\bar{\xi })\in \Lambda \) and \(\bar{\delta }\) is as in (2.6), then there holds
as \(\varepsilon \rightarrow 0\), \(C^0\)-uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\).
Proof
It’s easy to verify
where \(F_\varepsilon (u)=\int \limits _{0}^uf_\varepsilon (s)ds\), \(G_\varepsilon (u)=\int \limits _{0}^ug_\varepsilon (s)ds\). By the Hölder inequality, Proposition 3.1, Lemma 4.2, and (2.4), for any \(\vartheta \in (0,1)\), we get
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\). Moreover, by the mean value formula, Lemma 2.5, (5.14), (5.16) and the Sobolev embedding theorem, for any \(\vartheta \in (0,1)\), we obtain
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\). This ends the proof. \(\square \)
Next, we estimate the gradient of the reduced energy.
Lemma 5.4
Under the assumptions on p, q and N of Theorem 1.4, if \((\bar{\delta },\bar{\xi })\in \Lambda \) and \(\bar{\delta }\) is as in (2.6), then for any \(m=1,2,\cdots ,k\), there holds
and set \(\bar{\xi }(y)=\big (\exp _{\xi _1}(y^1),\exp _{\xi _2}(y^2),\cdots ,\exp _{\xi _k}(y^k)\big )\), \(y=(y^1,y^2,\cdots ,y^k)\in B(0,r)^k\), for any \(l=1,2,\cdots ,N\), it holds that
as \(\varepsilon \rightarrow 0\), \(C^0\)-uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\), where the function \(\Psi _k(\bar{t},\bar{\xi })\) is defined as (3.5).
Proof
For any \((\varphi ,\psi )\in \mathcal {X}_{p,q}(\mathcal {M})\), by Proposition 3.1, there exist \(c_{10},c_{11},\cdots ,c_{1N}\), \(c_{20},c_{21},\cdots ,c_{2N}\), \(\cdots \), \(c_{k0},c_{k1},\cdots ,c_{kN}\) such that
We claim that: for any \(\vartheta \in (0,1)\), there holds
Taking \((\varphi ,\psi )=(\Psi _{\delta _j,\xi _j}^i,\Phi ^i_{\delta _j,\xi _j})\), \(0\le i\le N\), \(1\le j\le k\), by (2.11)–(2.14), we have
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\), where \(\chi _{\delta _j}(x)=\chi ({\delta _j|x|})\). On the other hand, it follows from \((\Psi _{\varepsilon ,\bar{t},\bar{\xi }},\Phi _{\varepsilon ,\bar{t},\bar{\xi }})\in \mathcal {Z}_{\bar{\delta },\bar{\xi }}\) that
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\), where we have used the fact that \(\Vert \Psi ^i_{\delta _j,\xi _j}\Vert _{q+1}<+\infty \) and \(\Vert \Phi ^i_{\delta _j,\xi _j}\Vert _{p+1}<+\infty \) for any \(1<p\le \frac{N+2}{N-2}\le q\), \(i=0,1,\cdots ,N\) and \(j=1,2,\cdots ,k\). From (5.22) and (5.23), we prove the claim.
By (5.3) and (5.4), we can compute
and
as \(\varepsilon \rightarrow 0\). Next, we estimate (5.24) and (5.25). By the Hölder inequality, Proposition 3.1, and the Sobolev embedding theorem, arguing as Lemma 4.2, for any \(l=0,1,\cdots ,N\), we have
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\). Moreover, by the mean value formula, Lemma 2.5, (5.14), (5.16) and the Sobolev embedding theorem, for any \(l=0,1,\cdots ,N\), we obtain
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\). Using (2.4), (5.8), (5.9), (5.20) and (5.21), for any \(l=1,2,\cdots ,N\), we get
and
as \(\varepsilon \rightarrow 0\), uniformly with respect to \(\bar{\xi }\) in \(\mathcal {M}^k\) and to \(\bar{t}\) in compact subsets of \((\mathbb {R}^+)^k\). Taking \(\frac{3}{4}<\vartheta <1\), we complete the proof. \(\square \)
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The research has been supported by Natural Science Foundation of Chongqing, China CSTB2024NSCQ-LZX0038.
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Chen, W., Wang, Z. Multiple Blowing-Up Solutions for Asymptotically Critical Lane-Emden Systems on Riemannian Manifolds. J Geom Anal 34, 298 (2024). https://doi.org/10.1007/s12220-024-01722-6
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DOI: https://doi.org/10.1007/s12220-024-01722-6