Abstract
We are concerned with the Sinh-Gordon equation in bounded domains. We construct blow up solutions with residual mass exhibiting either partial or asymmetric blow up, i.e. where both the positive and negative part of the solution blow up. This is the first result concerning residual mass for the Sinh-Gordon equation showing in particular that the concentration-compactness theory with vanishing residuals of Brezis-Merle can not be extended to this class of problems.
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1 Introduction
We are concerned with the following Sinh-Gordon equation
where \(\Omega \subset {{\mathbb {R}}}^2\) is smooth and bounded and \(\rho ^+, \rho ^-\) are two positive parameters. The latter problem arises as a mean field equation in the study of the equilibrium turbulence [21, 24]. Moreover, it is also related to constant mean curvature surfaces [20, 29]. Observe that for \(\rho ^-=0\), (1.1) reduces to the standard Liouville equation which has been extensively studied in the literature. Therefore, many efforts have been done to study existence [3, 12,13,14,15] and blow up phenomena [1, 10, 16, 17, 20, 23, 25, 26, 28] for this class of problems.
In the present paper we further explore the blow up phenomenon of (1.1). Let \(u_n\) be a sequence of solutions to (1.1) corresponding to \(\rho ^{\pm }=\rho _n^{\pm }\le C\). Define the positive and negative blow up set as
We have \(S_{\pm }\cap \partial \Omega =\emptyset \) by [1], and \(S_{\pm }\subset \Omega \) is finite by the argument of [5]. For \(p\in S_{\pm }\) the local mass is defined by
By [16, 20] we know that \(m_{\pm }(p)\) satisfy a quantization property, i.e. \(m_{\pm }(p)\in 8\pi {\mathbb {N}}\). Moreover, in view of the relation
see for example [23], the couple \((m_+, m_-)\), up to the order, takes the value in the set
see [16, 20]. Finally, by standard analysis [23], one has, for \(n\rightarrow +\infty \),
in the sense of measures, where \(r_{\pm }\in L^1(\Omega )\) are residual terms. From the above convergence, \(\rho ^{\pm }\) will be called global masses of the blow up solutions. Observe that both the local masses and the residual terms affect the global masses. In striking contrast with the concentration-compactness theory of Brezis-Merle [5], the latter residuals may not be zero a priori. This fact has important effects in the blow up analysis, variational analysis and Leray-Schauder degree theory of (1.1). One of the goals of the present paper is to provide the first explicit example of blow up solutions exhibiting residual terms, thus confirming that the concentration-compactness theory can not be extended to this class of problems.
1.1 Partial blow up
We start here with a related problem, that is partial blow up with prescribed global mass. More precisely, we look for blowing up solutions \(-u_n\) with \(\rho ^-_n\rightarrow 8\pi k, k\in {\mathbb {N}}\), such that \(u_n\) have prescribed global mass \(\rho ^+_n=\rho ^+\in (0, 8\pi )\). To this end we introduce
and consider the following singular (at \(\xi _i\in \Omega \)) mean field equation:
where \({{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega \) and \(h(x,{{\varvec{\xi }}})=e^{-8\pi \sum _{i=1}^kG(x,\xi _i)}\). Here G(x, y) is the Green function of the Laplace operator in \(\Omega \) with Dirichlet boundary condition and we denote its regular part by H(x, y). Equation (1.4) is the Euler-Lagrange equation of the functional
To the latter functional and (a combination of) the Green functions we associate the following map:
It is known by [2] that if \(\Omega \) is simply connected and \(\rho ^+\in (0, 8\pi )\), then for any \({{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega \) there exists a unique solution to (1.4) and the solution is non-degenerate, in the sense that the linearized problem admits only the trivial solution. Then, by making use of the implicit function theorem it is not difficult to show that the function \(\Lambda \) is smooth, see for example [8]. Finally, as in [22], a compact set \({\mathcal {K}}\subset {\mathcal {F}}_k\Omega \) of critical points of \(\Lambda \) is said to be \(C^1\)-stable if, fixed a neighborhood \({\mathcal {U}}\) of \({\mathcal {K}}\), any map \(\Phi : {\mathcal {U}}\rightarrow {{\mathbb {R}}}\) sufficiently close to \(\Lambda \) in \(C^1\)-sense has a critical point in \({\mathcal {U}}\).
The first result of this paper is the following.
Theorem 1.1
Let \(\Omega \) be simply connected, \(\rho ^+\in (0, 8\pi )\) and let \({\mathcal {K}}\subset {\mathcal {F}}_k\Omega \), \(k\in {\mathbb {N}}\), be a \(C^1\)-stable set of critical points of \(\Lambda \). Then, there exists \(\lambda _0>0\) such that for any \(\lambda \in (0,\lambda _0)\) there exists a solution \(u_\lambda \) of (1.1) with \(\rho ^{\pm }=\rho _\lambda ^{\pm }\) such that the following two properties hold:
-
1.
\(\rho ^+_\lambda =\rho ^+ , \ \rho ^-_\lambda \rightarrow 8k\pi \) as \(\lambda \rightarrow 0\).
-
2.
There exist \({{\varvec{\xi }}}(\lambda )\in {\mathcal {F}}_k\Omega \) and \(\delta _i(\lambda )>0\) such that \(d({{\varvec{\xi }}}, {\mathcal {K}})\rightarrow 0, \ \delta _i\rightarrow 0\) and
$$\begin{aligned} u_\lambda (x)\rightarrow z(x,{{\varvec{\xi }}})-\sum _{i=1}^k\Big (\log \frac{1}{(\delta _i^2+|x-\xi _i|^2)^2}+8\pi H(x, \xi _i) \Big ) \quad \text{ in } H^1_0(\Omega ), \end{aligned}$$as \(\lambda \rightarrow 0\), where z solves (1.4) .
Some comments are in order. The assumptions that \(\Omega \) is simply connected and \(\rho ^+\in (0, 8\pi )\) guarantee the existence of a unique non-degenerate solution to (1.4): in general, the above result holds true whenever such solution exists. For example, one can drop the condition on \(\Omega \) by assuming \(\rho ^+\) to be sufficiently small, see for example [8].
On the other hand, if \(\Omega \) is simply connected and \(\rho ^+\in (0, 8\pi )\) it is not difficult to show that for \(k=1\) the minimum of \(\Lambda \) is a \(C^1\)-stable set of critical points of \(\Lambda \), see for example [8]. Moreover, for non-simply connected domains the function \(\Lambda \) always admits a \(C^1\)-stable set of critical points [7].
Therefore, the conclusion of Theorem 1.1 holds true if either \(\Omega \) is simply connected, \(\rho ^+\in (0, 8\pi )\) and \(k=1\), or \(\Omega \) is multiply connected, \(\rho ^+\) sufficiently small and \(k\ge 1\). Finally, the location of the blow up set can be determined by using the following expression, which can be derived similarly as in [8]:
1.2 Asymmetric blow up
We next construct blow up solutions with residual mass exhibiting the asymmetric blow up, i.e. where both the positive and negative part of the solution blow up. Since the local masses \((m_+,m_-)\) belong to the set \(\Sigma \) defined in (1.2), for \(k \ge 2\) we look for blowing up solution \(u_n\) with \(\rho _n^-\rightarrow 4\pi k(k+1)\) and \(\rho _n^+= \rho ^+=4\pi k(k-1)+\rho _0\), where \(\rho _0\in (0, 8\pi )\) is a fixed residual mass. For simplicity of presentation we assume that k is odd, the case of k even being similar. We consider here l-symmetric domains \(\Omega \) with \(l\ge 2\) even, i.e. if \(x\in \Omega \) then \({\mathcal {R}}_l\cdot x\in \Omega \), where
Consider then the following singular (at \(x=0\)) mean field equation:
Again by [2] we know that if \(\Omega \) is simply connected and \(\rho ^0\in (0, 8\pi )\), then there exists a unique non-degenerate solution to (1.8).
The second result of this paper is the following.
Theorem 1.2
Let \(\Omega \) be a simply connected l-symmetric domain according to (1.7) and \(\rho ^+=4\pi k(k-1)+\rho _0\) with \(k\in {\mathbb {N}}\) odd, \(l\ge 2\) even and \(\rho _0\in (0, 8\pi )\). Then, there exists \(\lambda _0>0\) such that for any \(\lambda \in (0,\lambda _0)\), there exists a solution \(u_\lambda \) of (1.1) with \(\rho ^{\pm }=\rho _\lambda ^{\pm }\) such that the following two properties hold:
-
1.
\( \rho ^+_\lambda =\rho ^+ , \ \rho ^-_\lambda \rightarrow 4\pi k(k+1)\) as \(\lambda \rightarrow 0\).
-
2.
There exists \(\delta _i(\lambda )\rightarrow 0\) (defined in (4.2)) such that
$$\begin{aligned} u_\lambda (x)\rightarrow z(x)+\sum _{i=1}^k(-1)^i\Big (\log \frac{1}{(\delta _i(\lambda )^{\alpha _i} +|x|^{\alpha _i})^2}+4\pi \alpha _iH(x, 0) \Big ) \quad \text{ in } H^1_0(\Omega ), \quad \alpha _i=4i-2, \end{aligned}$$as \(\lambda \rightarrow 0\), where z solves (1.8) .
Observe that the assumption that \(\Omega \) is simply connected and \(\rho _0\in (0, 8\pi )\) is used only to ensure the existence of a non-degenerate solution to (1.8): in general, the above result holds true whenever such solution exists. On the other hand, the symmetry condition of the domain is imposed to rule out the degeneracy of the singular Liouville equation.
The argument follows the strategy introduced in [8, 9] for the Toda system, that is a system of Liouville-type equations, and it is based on the perturbation method starting from an approximate solution and studying the invertibility of the linearized problem. The main difficulty is due to the coupling of the local and global nature of the problem since we are prescribing both the local and global masses. In particular, blow up solutions of (1.1) with local masses \((4\pi k(k-1), 4\pi k(k+1))\) have been constructed in [11] by superposing k different bubbles with alternating sign. Gluing the solution of (1.8) to the latter blow up solutions we are able to construct blow up solutions with residual mass, that is with \(\rho ^+_n=\rho ^+=4\pi k(k-1)+\rho _0\) and \(\rho ^-_n\rightarrow 4\pi k(k+1)\) for any \(k\ge 2\). In this generality the latter construction is quite delicate and technically more difficult compared to the one in [9, 11], since we have more sign-changing singular bubbles, and since one need to consider all the interactions of different bubbles and also the interaction with the global solution z(x) which makes the linear theory more complicated, see the linear theory in Sect. 4.3. We remark that the same strategy can be carried out for more general asymmetric Sinh-Gordon equations, for example for the Tzitzéica equation [18].
The paper is organized as follows. Sect. 2 contains some notation and preliminary results which will be used in the paper. Sect. 3 is devoted to the proof of Theorem 1.1 while the proof of Theorem 1.2 is derived in Sect. 4.
2 Preliminaries
In this section we collect some notation and useful information that we will use in this paper. We shall write
to denote the norm in \(H_0^1(\Omega )\) and in \(L^p(\Omega )\), respectively, for \(1\le p\le +\infty \). For \(\alpha \ge 2\), let us define the Hilbert spaces:
with \(\Vert u\Vert _{L_{\alpha }}\) and \(\Vert u\Vert _{H_{\alpha }}:=(\Vert \nabla u\Vert _{L^2({{\mathbb {R}}}^2)}^2+\Vert u\Vert _{L_{\alpha }}^2)^{\frac{1}{2}}\) denoting their norms, respectively. For simplicity, we will denote \(L_2\) and \(H_2\) by L and H, respectively. Let us recall that the embedding \(H_\alpha ({{\mathbb {R}}}^2)\rightarrow L_\alpha ({{\mathbb {R}}}^2)\) is compact [11]. For any \(p>1\), let \(i_p^*: L^p(\Omega )\rightarrow H_0^1(\Omega )\) be the adjoint operator of the embedding \(i_p: H_0^1(\Omega )\rightarrow L^{\frac{p}{p-1}}(\Omega )\), i.e. for \(v\in L^{p}(\Omega )\) , \(u=i_p^*(v)\) if and only if in the weak sense
Then one has \(\Vert i_p^*(v)\Vert _{H_0^1(\Omega )}\le c_p \Vert v\Vert _p\) for some constant \(c_p>0\) depending only on \(\Omega \) and \(p>1\).
The symbol \(B_r(p)\) will stand for the open metric ball of radius r and center p. To simplify the notation we will write \(B_r\) for balls which are centered at 0. Throughout the whole paper c, C will stand for constants which are allowed to vary among different formulas or even within the same line.
3 Partial blow up
3.1 Approximate solutions
In order to prove Theorem 1.1 we introduce the associated equation
where \(\lambda >0 \) will be suitably chosen small. By the definition of \(i_p^*\), problem (3.1) is equivalent to the following:
where \(F(u)=\rho ^+g(u)-\lambda f(u)\) and
First let us introduce the approximate solutions we will use. Recall that solutions of the following regular Liouville equation [6]:
are given by
for \(\delta >0, \ \xi \in {{\mathbb {R}}}^2\) and we set
Since we are considering Dirichlet boundary condition, let us introduce the projection:
By the maximum principle,
where H(x, y) is the regular part of the Green’s function of the Dirichlet Laplacian in \(\Omega \), \(G(x,y)=\frac{1}{2\pi }\log \frac{1}{|x-y|}+H(x,y)\), see [11] for the derivation of (3.4)
Let \(k\ge 1\), fix \({{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega \) and consider \(z(x, {{\varvec{\xi }}})\) which is the unique solution to (1.4). The approximate solutions we will use are given by
where the parameters \(\delta _i\) are suitably chosen such that
Our aim is to find a solution u to (3.1) of the form \(u=W+\phi \) where \(\phi \) is small in some sense. Before we go further, let us first collect some useful well-known facts.
As it is shown in [4], any solution \(\psi \in H\) of
can be expressed as a linear combination of
Moreover, the projections of \(Z_{\delta , \xi }^i\) have the following expansion:
Finally, by straightforward computations and taking into account the choice of \(\lambda \) in (3.6) the following estimates hold true [8]:
Lemma 3.1
For any \({\mathcal {C}}\subset {\mathcal {F}}_k\Omega \) compact and \({{\varvec{\xi }}}\in {\mathcal {C}}\), one has
and there exists some \(a>0\) such that for any \(i=1,\cdots ,k\) and \(j=1,2\), it holds that
and
where \(Z_i^j=Z_{\delta _i,\xi _i}^j\) and \(\langle u,v\rangle =\int _{\Omega }\nabla u\cdot \nabla v\, dx\).
In the section, we set
and
Denote by
be the corresponding projections. To solve (3.1), it is equivalent to solve the following system:
3.2 Estimate of the error
We next estimate the error of the approximate solution:
Lemma 3.2
For any \(p\ge 1\) we have, for \({{\varvec{\xi }}}\in {\mathcal {C}}\subset {\mathcal {F}}_k\Omega \), \({\mathcal {C}}\) compact,
Moreover, \(\Vert {\mathcal {R}}\Vert \le c_p\Vert R\Vert _p\) for some \(c_p>0\) depending on \(p, \Omega \).
Proof
By the definition of W,
Estimate of \(E_1=\Big (\sum _i e^{w_i}-\lambda e^{\sum _i Pw_i-z(x,{{\varvec{\xi }}})}\Big )\). Take \(\eta >0\) such that \(|\xi _i-\xi _j|\ge 2\eta \) and \(d(\xi _i,\partial \Omega )\ge 2\eta \). First, using (3.4), we have
Hence, on \(B_{\eta }(\xi _i)\), writing \(x=\xi _i+\delta _i y\), one has
Thus
It follows that
Moreover,
Combining the above estimates,
Estimate of \(E_2=\Big (\Delta z(x,{{\varvec{\xi }}})+\rho ^+\frac{e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}{\int _\Omega e^{z(x,{{\varvec{\xi }}})-\sum _i Pw_i}}\Big )\). First of all,
where
So
One has
since \(z(x,{{\varvec{\xi }}})\) is a solution of (1.4). Thus
Estimate of \(\partial _{{\varvec{\xi }}}E_1\). Next we consider the derivatives. By straightforward computations we get
It is then not difficult to show that
Combining all the above estimates,
Estimate of \(\partial _{{\varvec{\xi }}}E_2\). The estimate of the derivative of \(E_2\) is analogous. Using the equation satisfied by \(z(x,{{\varvec{\xi }}})\) in (1.4) and (3.15),
Thus we have
Finally, combining the estimates for \(E_1\) and \(E_2\), we have
Once we get the estimate for R, the estimate for \({\mathcal {R}} \) follows directly. \(\square \)
3.3 The linear operator
In this subsection, we consider the following problem: given \(h\in H_0^1(\Omega )\) we look for a function \(\phi \in H_0^1(\Omega )\) and \(c_{ij}\) such that
It is equivalent to
where
Let \(L: K^\perp \rightarrow K^\perp \) be the linear operator defined by
then the problem is equivalent to first solving \(\phi \) for
and then finding \(c_{ij}\) for
First we have the following apriori estimate:
Lemma 3.3
Let \({\mathcal {C}}\subset {\mathcal {F}}_k\Omega \) be a fixed compact set. Then, there exist \(\lambda _0>0\) and \(C>0\) such that for any \(\lambda \in (0,\lambda _0)\), \({{\varvec{\xi }}}\in {\mathcal {C}}\) and \(h\in H_0^1(\Omega )\), any solution \(\phi \in H_0^1(\Omega )\) of
satisfies
Proof
We prove it by contradiction. Assume there exist \(\lambda _n\rightarrow 0\), \({{\varvec{\xi }}}_n\rightarrow {{\varvec{\xi }}}^*\in {\mathcal {F}}_k \Omega \), \(h_n\in H_0^1(\Omega )\) and \(\phi _n \in H_0^1(\Omega )\) which solves (3.24) with
For \(i=1, \cdots , k\), define \({\tilde{\phi }}_i(y)\) as
In the following, we omit the index n for simplicity.
Step 1. We claim that
and
Let \(\psi \in C_0^\infty (\Omega \setminus \{\xi ^*_1, \cdots ,\xi _k^*\})\), multiply equation (3.24) by \(\psi \) and integrate, then
By the assumption on \(\phi \), using the fact that in \(\Omega \setminus \{\xi _1^*,\cdots , \xi _k^*\}, \ e^{w_i}=O(\lambda )\) and \(e^W=h(x, {{\varvec{\xi }}})e^{z(x,{{\varvec{\xi }}})}+O(\lambda )\), one has
which gives
So \(\Vert \phi ^*\Vert _{H_0^1(\Omega )}\le 1\) and it solves
By the non-degeneracy of \(z(x, {{\varvec{\xi }}})\), we can get that \(\phi ^*=0\). Thus (3.26) is proved.
Now let us prove (3.25). Multiplying (3.24) again by \(\phi \) and integrating,
From the above equation, one can get that
where we used (3.26). So we get that \({\tilde{\phi }}_i\) is bounded in \(H({{\mathbb {R}}}^2)\). There exists \({\tilde{\phi }}_0\) such that
Let \({\tilde{\psi }}\in C_0^\infty ({{\mathbb {R}}}^2)\) and define \(\psi _i={\tilde{\psi }}(\frac{x-\xi _i}{\delta _i})\). Multiplying (3.24) by \(\psi _i\) and integrating over \(\Omega \),
Since \(\psi _i(x)=0\) if \(|x-\xi _i|\ge R\delta _i\) for some \(R>0\), we have
Passing to the limit in (3.27), we have
Moreover, by the orthogonality condition in (3.24), we have
So we deduce that
Step 2. We claim that \(\gamma _i=0\) for \(i=1,\cdots ,k\). Multiplying equation (3.24) by \(PZ_i^0\) and integrate over \(\Omega \),
Since
where \(Z^0=\frac{1-|y|^2}{1+|y|^2}\) and by (3.7),
for some \(p>1\), by Hölder inequality. Moreover, by (3.26), (3.7) and (3.15), one has
From (3.28) and the above estimates, one has
Next we multiply equation (3.24) by \(Pw_i\) and integrate over \(\Omega \),
Now we estimate the above equation term by term.
by (3.25) and the fact that
By the expansion of \(Pw_i\),
Moreover,
and
Combining all the above estimates, we have
which implies that \(\gamma _i=0\) since
Step 3. Finally, we derive a contradiction.
Multiply equation (3.24 ) by \(\phi \) and integrate:
From the estimates in step 1-2 and the assumptions on \(\phi \) and h, it is not difficult to show that the left hand side of the above equation tends to 1, while the right hand side has limit 0. This is a contradiction which concludes the proof. \(\square \)
Now we can derive a priori estimates for problem (3.20).
Proposition 3.4
Let \({\mathcal {C}}\subset {\mathcal {F}}_k\Omega \) be a compact set. Then, there exist \(\lambda _0>0\) and \(C>0\) such that for any \(\lambda \in (0, \lambda _0)\), \({{\varvec{\xi }}}\in {\mathcal {C}}\) and \(h\in H_0^1(\Omega )\), if \((\phi , c_{ij})\) is a solution of (3.20), we have
Proof
By Lemma 3.3 and (3.8) , we know that
In order to estimate \(c_{ij}\), multiply the equation (3.20) by \(PZ_i^j\) and integrating over \(\Omega \),
where in the last line we use (3.9). Since for any \(q\ge 1\),
we have
Summing all \(|c_{ij}|\) up and choosing suitable \(q\in (1, 2)\), we can get that
\(\square \)
From the above a priori estimate and the Fredholm alternative it is then standard to derive the following existence result.
Proposition 3.5
Let \({\mathcal {C}}\subset {\mathcal {F}}_k\Omega \) be a compact set. Then, there exist \(\lambda _0>0\) and \(C>0\) such that for any \(\lambda \in (0, \lambda _0)\), \({{\varvec{\xi }}}\in {\mathcal {C}}\) and \(h\in H^1_0(\Omega )\), there exists a unique solution \((\phi , c_{ij})\) of (3.20), which satisfies
Proof
For the first equation (3.22), since \(\phi \rightarrow \Pi ^\perp (i_p^*(M(W)[\phi ]))\) is the compact operator in \(K^\perp \), the existence and uniqueness of \(\phi \) follows from the Fredholm alternative and the above a priori estimate, and then \(c_{ij} \) are determined by (3.23). Once existence of \((\phi , c_{ij})\) is obtained, the estimate follows from Proposition 3.4. \(\square \)
3.4 Nonlinear problem
The aim of this subsection is to find \((\phi , \{c_{ij}\})\) such that \(u=W_{\xi }+\phi _{\xi }\) solves
For this purpose, we shall find a solution \(\phi \) of
where R is the error term defined in Sect. 3.2,
From the above linear theory, the existence of a solution to the nonlinear problem (3.30) follows a standard strategy using contraction mapping.
Proposition 3.6
Let \({\mathcal {C}}\subset {\mathcal {F}}_k\Omega \) be compact set. For any \(\epsilon >0\) sufficiently small, there exist \(\lambda _0>0\) and \(C>0\) such that for any \(\lambda \in (0, \lambda _0)\) and \({{\varvec{\xi }}}\in {\mathcal {C}}\), there exists a unique \((\phi , c_{ij})\) solution of (3.30) satisfying the estimates:
Proof
Denote the solution to (3.20) by \(\phi :=T(h)\). Then (3.30) is equivalent to
The solution \(\phi \) can be obtained through contraction mapping. Define
for \(\Lambda \) large and \(\lambda \) small and p close to 1.
From Proposition 3.5 and the error estimate for R, for \(\phi , \phi _1, \phi _2\in {\mathcal {B}}\), similarly to the estimate in Proposition 4.10 in [8], one has
So \({\mathcal {T}}\) maps \({\mathcal {B}}\) into itself and it is a contraction mapping. For \(\epsilon \) small, we can choose p sufficiently close to 1, such that \({\mathcal {B}}\subset \{\phi , \, \Vert \phi \Vert \le C\lambda ^{\frac{1}{2}-\epsilon }\}\}\). Since \({\mathcal {T}}\) is a contraction mapping in \({\mathcal {B}}\), we can also get that the fixed point in \({\mathcal {B}}\) is unique, i.e. the solution \(\phi \) is unique in \({\mathcal {B}}\). The estimate for \(\phi \) follows from the above estimates. The estimates for \(\partial _{\xi _i}\phi \) are obtained similarly to Proposition 4.10 in [8] and the estimate for \(c_{ij}\) follows from Proposition 3.4. \(\square \)
3.5 The reduced problem
We introduce here the finite-dimensional reduction. In the previous subsection we have found a solution \(u=W+\phi \) to the problem
Consider now the associated energy functional:
and let \({\tilde{J}}({{\varvec{\xi }}})=J(W_{{\varvec{\xi }}}+\phi _{{\varvec{\xi }}})\).
Lemma 3.7
Let \({{\varvec{\xi }}}\in {\mathcal {F}}_k\Omega \) be a critical point of \({\tilde{J}}\), then for \(\lambda \) small, \(u=W_{{{\varvec{\xi }}}}+\phi _{{{\varvec{\xi }}}}\) is a solution of (3.1).
Proof
If \({{\varvec{\xi }}}\) is a critical point of \({\tilde{J}}({\varvec{\xi }})\), then one has
which is equivalent to
Let us fix \(q>1\). Since
combining the estimate (3.31), one has
Combining the estimates (3.35) and (3.36), we conclude that
which implies that all \(c_{ij}\) are zero. So the corresponding u is a solution of (3.1) as desired. \(\square \)
Recall the definition of \(\Lambda \) in (1.5). We next consider the expansion of the energy.
Proposition 3.8
It holds
\({\mathcal {C}}^1\) uniformly in \({{\varvec{\xi }}}\) in compact sets of \(\Omega \).
Proof
By the definition of J(W) and W, one has
Using (3.16),
While using (3.13) and the estimate for \(E_1\),
where \({\tilde{\Omega }}_i=(\Omega -\xi _i)/\delta _i\). Moreover, using the expansion (3.4)
and for \(i\ne j\),
Combining all the above estimates, we have
Next, we consider the derivative of J(W).
where \(E_1, E_2\) were introduced in Lemma 3.2 and where we used
Using the definition of \(w_i\) and \(Z_i^j\), for \(\ell \ne i\)
Moreover, taking \(\eta >0\) such that \(|\xi _i-\xi _j|\ge 2\eta \) and \(d(\xi _i,\partial \Omega )\ge 2\eta \), we have
Let
Then,
Finally,
Combining the above estimates, we have
as desired, where we used (1.6). \(\square \)
Finally, we have the following expansion of the reduced energy.
Proposition 3.9
It holds
\({\mathcal {C}}^1\) uniformly in \({{\varvec{\xi }}}\) in compact sets of \({\mathcal {F}}_k\Omega \).
Proof
To simplify the notation, we shall drop the sub-index \({{\varvec{\xi }}}\) in the proof. It is not difficult to show that
Next we consider the derivatives.
Using the estimate for \(c_{ij}\) in Proposition 3.6 and (3.34), we have
provided q is sufficiently close to 1. Recalling the definitions of f, g in (3.3) we exploit now the estimates in [8, Lemma 4.7]. For some \(\theta \in (0,1)\) and p sufficiently close to 1 we have
Moreover, for some \({{\tilde{\theta }}}\in (0,1)\) and suitable p, q
Recall that
for \({{\varvec{\xi }}}\) in compact sets of \({\mathcal {F}}_k\Omega \). Then
by the orthogonality condition satisfied by \(\phi \). Moreover, again by the orthogonality condition we have
Combining the above estimates, we have
as desired. \(\square \)
Proof of Theorem 1.1
Let \({\mathcal {K}}\subset {\mathcal {F}}_k\Omega \) be a \(C^1\)-stable set of critical points of \(\Lambda \). Then, by Propositions 3.8-3.9, for \(\lambda >0\) small, there exists \({{\varvec{\xi }}}_\lambda \) critical point of \({\tilde{J}}\) and \(d({{\varvec{\xi }}}_\lambda , {\mathcal {K}})\rightarrow 0\) as \(\lambda \rightarrow 0\). By Lemma 3.7, \(u_\lambda =W_{\xi _\lambda }+\phi _{\xi _\lambda }\) is a solution of (3.1). It follows that \(u_\lambda \) solves the original problem (1.1) with \(\rho ^+_\lambda =\rho ^+\) and
Moreover, from the definition of \(u_\lambda \), and using (3.4), (3.5) and (3.31), we can derive the second property in Theorem 1.1. \(\square \)
4 Asymmetric blow up
4.1 Approximate solutions
In this section we will derive the proof of Theorem 1.2. To this end we will always assume that \(\Omega \) is \(l-\)symmetric for \(l\ge 2\) even according to (4.1) below. Therefore, we will consider symmetric functions such that
recall (1.7), and define
Consider problem (3.1) and let \(k\ge 2\) be an odd integer. In order to construct blow up solutions with local masses \((4\pi k(k-1), 4\pi k(k+1))\), we need to consider the following singular Liouville equation. Let \(\alpha \ge 2\). It is known [27] that
solves the problem
and
Similarly to the previous section, let Pu be the projection of the function u into \(H_0^1(\Omega )\). We look here for a sign changing solution of the form
where \(\phi \) is a small error term, z(x) is the unique solution of (1.8) and \(Pw_i=Pw_{\delta _i}^{\alpha _i}\) with
The latter parameters are chosen such that the interaction of different bubbles is small. More precisely, the following functions will play an important role in the interaction estimate:
where
As we will see in the sequel, in order to make these two functions small, we will need to choose \(\delta _i\) and \(\alpha _i\) such that
and
where \(h_i(x)=4\pi \alpha _iH(x,0).\) From (4.6) we deduce that \(\alpha _1=2\) and \(\alpha _i=\alpha _{i-1}+4\) for \(i\ge 2\) which implies the choice of \(\alpha _i\) in (4.2). On the other hand, from (4.7) and (4.8) one easily deduces that
and
From the above identities, one can get that
for some \(d_i>0\), which implies (4.2).
We estimate now \(\Theta _i\) and \(T_i\). First, using the maximum principle it is not difficult to see that
and for \(i,j=1,\cdots ,k\),
where \(h_i(x)=4\pi \alpha _iH(x,0)\).
Remark 4.1
From the above expansion, one can get that for \(|x|\ge \delta _0\) for \(\delta _0>0\) small, the following expansion holds:
From the definition of \(\alpha _i\) we have \(\sum _{i=1}^k(-1)^i\alpha _i=(-1)^k2k\) and hence, for k odd it holds
We next introduce the following shrinking annulus
where \(\delta _0:=0\) and \(\delta _{k+1}:=+\infty \).
Lemma 4.2
For any \(y\in \frac{A_i}{\delta _i}\), the following estimates hold:
In particular,
Proof
Consider \(y\in \frac{A_i}{\delta _i}\). From (4.10), and using (4.6) and (4.7), for i odd,
Similarly, for i even,
Finally, (4.14) follows from the above two estimates since \(\delta _i|y|=O(1)\) when \(y\in \frac{A_i}{\delta _i}\). \(\square \)
Finally, we will need the following non-degeneracy result for entire singular Liouville equations which was derived in [11, Theorem 6.1] for \(l=2\) and which can be extended to any \(l\ge 2\) even.
Proposition 4.3
Assume \(\phi : {{\mathbb {R}}}^2\rightarrow {{\mathbb {R}}}\) satisfying (4.1) is a solutions of
with \(\alpha \ge 2\) and \(\frac{\alpha }{2}\) odd. Then,
4.2 Estimate of the error term
In this subsection we estimate the error of the approximate solution. To this end, set
Lemma 4.4
For any \(q\ge 1\) sufficiently close to 1, the following holds:
Proof
First we consider \(E_2\). Recall the definition of the annulus \(A_i\) in (4.11).
One has
Let us estimate \(I_{11}\). For fixed i odd,
provided that q is close to 1. Therefore, we get \(I_{11}=O(\lambda ^{\frac{2-q}{2(2k-1)}}).\)
For \(I_{12}\), fix \(j\ne i\) odd,
provided that q is close to 1. Therefore, \(\Vert I_1\Vert _q=O\Big (\lambda ^{\frac{2-q}{2q(2k-1)}}\Big )\).
Next, let us estimate \(I_2\). For l even fixed,
We have,
if q is close to 1. Moreover, similarly to the estimate of \(I_{12}\), one can also get that \(I_{22}=O\Big (\lambda ^{\frac{2-q}{2(2k-1)}}\Big ).\)
Combining all the above estimates, one has
Next we consider \(E_1\). First we need to estimate \(\int _{\Omega }e^{W}dx\). For i even fixed,
where we have used Lemma 4.2 for the estimate of \(T_i(y)\) and the fact that
For \(i<k\) odd and fixed, reasoning as in (4.16) with \(q=1\), one has
Finally for \(i=k\) which is odd, using Remark 4.1,
In conclusion, one has
where we used the definition of Q in (4.5) and the fact that
since \(\sum _{i\ even}4\pi \alpha _i=4\pi k(k-1)=\rho ^+-\rho _0\).
With the estimate for \(\int _{\Omega }e^Wdx\) in hand, we now consider \(E_1\).
First for i even fixed,
So we have
Next, consider \(J_2\). For \(l<k\) odd and fixed, similarly to the estimates in (4.16), (4.15) and using (4.18)
Finally, we consider the case \(l=k\) which is odd: using (4.18) and (4.15)
In conclusion, one has
\(\square \)
4.3 The linear theory
In this subsection, we consider the linear problem: given \(h\in {\mathcal {H}}_l\), we look for \(\phi \in {\mathcal {H}}_l\) such that
First we have the following apriori estimate:
Lemma 4.5
There exist \(\lambda _0>0\) and \(C>0\) such that for any \(\lambda \in (0,\lambda _0)\), \(h\in {\mathcal {H}}_l\) and \(\phi \in {\mathcal {H}}_l\) solution of (4.20) we have
We start by listing some straightforward integrals which will be useful in the proof of Lemma 4.5.
Lemma 4.6
The following hold:
Proof of Lemma 4.5
We prove it by contradiction. Assume there exist \(\lambda _n\rightarrow 0\), \(h_n\in {\mathcal {H}}_l\) and \(\phi _n \in {\mathcal {H}}_l\) which solves (4.20) such that
In the following, we omit the index n for simplicity. For \(i=1, \cdots , k\), define \({\tilde{\phi }}_i(y)\) as
Step 1. We claim that
and
Letting \(\psi \in C_0^\infty (\Omega \setminus \{0\})\) and multiplying equation (4.20) by \(\psi \) and integrating, one has
By the assumption on \(\phi \), using the fact that in compact sets of \(\Omega \setminus \{0\}\),
one has
where
So \(\Vert \phi ^*\Vert _{H_0^1(\Omega )}\le 1\) and it solves
By the non-degeneracy of z(x) we get \(\phi ^*=0\). Thus (4.24) is proved.
Now we prove that \({\tilde{\phi }}_i\) is bounded in \(H_{\alpha _i}({{\mathbb {R}}}^2)\). First it is easy to check that
We multiply (4.20) again by \(\phi \) and integrate,
From the above equation, one can get that,
where we used (4.24). Let i be odd. Lemma 4.4 gives
or equivalently
Combined with (4.26), we deduce that \({\tilde{\phi }}_i\) is bounded in \(H_{\alpha _i}({{\mathbb {R}}}^2)\) when i is odd.
We consider now the case for i even. From (4.18), \(e^W=e^{z-8k\pi G(x,0)}+O(\lambda )\) uniformly on compact sets of \(\Omega \setminus \{0\}\) and recalling (4.24), we get that
Moreover, by (4.27) one can get that
Combining (4.28) and (4.29), we have
By Lemma 4.4, (4.24) and (4.30), \(\int _{\Omega }|x|^{\alpha _i-2}e^{w_i}\phi ^2dx=O(1)\) for i even, which implies that
So we get that also for i even, \({\tilde{\phi }}_i\) is bounded in \(H_{\alpha _i}({{\mathbb {R}}}^2)\).
Step 2. We claim that
From Step 1, we know that \({\tilde{\phi }}_i\rightarrow {\tilde{\phi }}_i^*\) weakly in \(H_{\alpha _i}({{\mathbb {R}}}^2)\) and strongly in \(L_{\alpha _i}({{\mathbb {R}}}^2)\). Consider \({\tilde{\psi }}\in C_0^\infty ({{\mathbb {R}}}^2\setminus \{0\})\) and let \({\mathcal {K}}\) be its support. For n large, one has
Define \(\psi _i={\tilde{\psi }}(\frac{x}{\delta _i})\). Multiplying (4.20) by \(\psi _i\) and integrating over \(\Omega \),
Consider first i even. According to Lemma 4.4, one has
where in the last line we used (4.24). Similarly, one has
Thus, \({\tilde{\phi }}_i^*\) satisfies
From this we deduce that the function
is a solution of
Since \(\int |\nabla {\tilde{\phi }}_i^*|^2dy\le 1\), \({\tilde{\phi }}_i^*\) is a solution in the whole space \({{\mathbb {R}}}^2\). By Proposition 4.3, we get that \({\tilde{\phi }}_i^*-\frac{1}{\rho ^+}\int _{{{\mathbb {R}}}^2}\frac{2\alpha _i^2|y|^{\alpha _i-2}}{(1+|y|^{\alpha _i})^2}{\tilde{\phi }}_i^*dy =\gamma _i \frac{1-|y|^{\alpha _i}}{1+|y|^{\alpha _i}}\) for some \(\gamma _i\). By(4.21) one has
which implies that
Since \(\rho ^+\ne 4\pi \alpha _i\) we deduce that
Hence, (4.31) is proved for i even.
We next turn to i odd. In this case, we consider (4.32) with i odd and estimate each term separately,
and
Hence, \({\tilde{\phi }}_i^*\) satisfies
namely \({\tilde{\phi }}_i^*\) is a solution of
and again we conclude by using Proposition 4.3.
Step 3. In this step, we will prove some estimates on the speed of convergence. We set
We will show that
Set \(Z_i^0=\frac{\delta _i^{\alpha _i}-|x|^{\alpha _i}}{\delta _i^{\alpha _i}+|x|^{\alpha _i}}\), we know that \(Z_i^0\) is a solution of
Let \(PZ_i^0\) be its the projection onto \(H_0^1(\Omega )\), that is
By maximum principle one can show
which implies
and
First we consider i even. Multiply (4.20) by \(PZ_i^0\) and integrate over \(\Omega \),
For the first term,
By Lemma 4.4, (4.24), (4.35) and (4.37),
For \(j\ne i\),
where we used (4.36).
Replace \(\phi \) by 1 in the estimate of (4.41) , one has for \(j\ne i\),
Using Lemma 4.4 , (4.42) and (4.37),
Moreover,
and again by Lemma 4.4 and (4.41)
Finally, for the last term,
Combining (4.38), (4.41), (4.39), (4.40), (4.43), (4.44), (4.45) and (4.46), we deduce that for i even,
Next we consider (4.38) for i odd. In this case, again we estimate (4.38) term by term. Similarly to the estimate for i even, first by Lemma 4.4, (4.37) and (4.41), one has
and
Combining all these terms, one can get that for i odd,
By considering the difference of (4.47) and (4.50), one has the following:
From (4.50), we first have \(\sigma _1(\lambda )=o(1)\). From (4.51), we have
Step 4. We claim that \(\gamma _i=0\) for \(i=1,\cdots ,k\).
When i is even, multiplying equation (4.20) by \(Pw_i\) and integrating over \(\Omega \),
Now we estimate the above equation term by term. For the first term, we have
To estimate the second term, by Lemma 4.4 and (4.24), we have
Based on (4.56), by the definition of \(\sigma _j(\lambda )\) (see (4.34)), Lemma 4.6 and (4.31), we get
where we used [11, (4.18)-(4.20) ].
Then by (4.57) and (4.55), one has
Similarly, by replacing \(\phi \) by 1 in (4.57), one can deduce that
and the estimate for \( \rho ^+\frac{\int _{\Omega }e^W\phi dx}{\int _{\Omega }e^W dx} \) has been obtained in (4.48).
Moreover,
and
Putting all the estimates in (4.54), (4.58), (4.59), (4.48), (4.60) and (4.61) into (4.53), we get that for i even,
Next we consider i odd. Similarly to the previous estimates, one has
and
Putting all the estimates in (4.54), (4.63), (4.64), (4.48), (4.65) and (4.61) into (4.53), we have for i odd,
Combining (4.52), (4.62) and (4.66), we have
from which we deduce that \(\gamma _i=0\) for \(i=1,\cdots ,k\).
Step 5. Finally, we derive a contradiction.
Multiplying equation (4.20) by \(\phi \) and integrating, we get
From (4.24) and the assumptions on \(\phi \) and h, we have that the left hand side of the above equation tends to 1 while the right hand side is of order o(1). This yields a contradiction. \(\square \)
Using the a priori estimates in Lemma 4.5 and the Fredholm alternative, we have the following existence result similarly to the proof of Proposition 3.5:
Proposition 4.7
There exist \(\lambda _0>0\) and \(C>0\) such that for any \(\lambda \in (0,\lambda _0)\), \(h\in {\mathcal {H}}_l\) there exists a unique solution \(\phi \in {\mathcal {H}}_l\) solution of (4.20) satisfying
4.4 Conclusion
By exploiting the linear theory developed in the previous subsection it is then standard to derive an existence result for the nonlinear problem (4.67) based on the contraction mapping, similarly to Proposition 3.6. We here give the sketch proof.
Proposition 4.8
For any \(\epsilon >0\) sufficiently small, there exist \(\lambda _0>0\) and \(C>0\) such that for any \(\lambda \in (0, \lambda _0)\), there exists a unique \(\phi \in {\mathcal {H}}_l\) solution of
satisfying
Proof
The proof is similar to Proposition 3.6 in the last section. The nonlinear problem (4.67) is equivalent to
where
Denote the solution to (4.20) by \(\phi :={\bar{T}}(h)\), then (4.69) is equivalent to
Define
for \(\Lambda \) large and \(\lambda \) small. If we choose p sufficiently close to 1, one can see that \(\bar{{\mathcal {B}}}\subset \{\phi , \, \Vert \phi \Vert \le C\lambda ^{\frac{1}{2(2k-1)}-\epsilon } \}\}\).
From Proposition 4.7 and the error estimate for \({\bar{E}}\), for \(\phi , \phi _1, \phi _2\in \bar{{\mathcal {B}}}\), similarly to the estimate in Proposition 5.4 in [9], one has
So \(\bar{{\mathcal {T}}}\) maps \(\bar{{\mathcal {B}}}\) into itself and it is a contraction mapping, we can get that the solution \(\phi \) is unique in \(\bar{{\mathcal {B}}}\). The estimate for \(\phi \) follows from the above estimate. \(\square \)
Proof of Theorem 1.2
By Proposition 4.8, \(u_\lambda =W_{\lambda }+\phi _\lambda \) is a solution to the original problem (1.1) with \(\rho ^+_\lambda =\rho ^+=4\pi k(k-1)+\rho _0\) and \(\rho ^-_{\lambda }=\lambda \int _{\Omega }e^{-u_\lambda }dx\). Then by Lemma 4.4 and (4.68)
Moreover, from the definition of \(u_\lambda \) and properties (3.37), (4.68) and (4.9), the second property in Theorem 1.2 can be derived easily. \(\square \)
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Ao, W., Jevnikar, A. & Yang, W. Blow up solutions for Sinh-Gordon equation with residual mass. Calc. Var. 61, 209 (2022). https://doi.org/10.1007/s00526-022-02317-1
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DOI: https://doi.org/10.1007/s00526-022-02317-1