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The blow up analysis of solutions of the elliptic sinh-Gordon equation. (English) Zbl 1137.35061

In this paper, the authors provide the relationship, in particular the blow-up analysis, between the analytic aspects of the elliptic sinh-Gordon equation on a two-dimensional surface and its geometric interpretation in terms of constant mean curvature surfaces and harmonic maps. The sinh-Gordon equation of the form
\[ u_{z\bar z} + \lambda \sinh z = 0 \tag{1} \]
plays an important role in the study of the construction of constant mean curvature surfaces and it also arises from many mathematical and physical problems. The authors investigate the blow up analysis of the solutions to (1) on a Riemann surface or on a bounded smooth domain in \({\mathbb R}^2\) and give precise asymptotic behavior when the sequence of solutions blows up as \(\lambda_n \to \lambda\). Let \(v_n\) be a sequence of solutions of the equation
\[ \Delta v_n = \lambda_n (e^{v_n} - e^{v_n})\tag{2} \]
on a Riemann surface \(\Sigma\) with the condition
\[ \int_{\Sigma} \lambda_n (e^{v_n} + e^{- v_n}) \,dv_g \leq C < \infty, \] and \(\lim_{n\to \infty} \lambda_n = \lambda\). Let \(p\in \Sigma\) be a point such that there exits a sequence \(x_n\) tending to \(p\) and \(v_n(x_n) \to \infty\), or \(-v_n(x_n) \to \infty\) and set
\[ m_1(p) = \lim_{r\to 0}\lim_{n\to \infty} \int_{B_r(p)} \lambda_n e^{v_n}\,dv_g \]
and
\[ m_2(p) = \lim_{r\to 0}\lim_{n\to \infty} \int_{B_r(p)} \lambda_n e^{- v_n}\,dv_g. \]
The main theorem of this paper states that the blow up values \(m_1\) and \(m_2\) are multiples of \(8 \pi\). As a corollary with a relationship \((m_1(p)-m_2(p))^2 = 8\pi (m_1(p)+m_2(p))\), it is obtained that the blow up values of the sinh-Gordon equation (2) can only be
\[ (m_1(p)), m_2(p)) = 8\pi \left(\frac{l(l-1)}{2}, \frac{l(l+1)}{2}\right) \quad \text{or}\quad \left(\frac{l(l+1)}{2}, \frac{l(l-1)}{2}\right) \]
for some integer \(l\geq 0\). The question arises, whether the constant \(l\) has any other value than one. Using the main results, the authors prove an existence result for the following problem as a byproduct:
\[ \Delta v_n = \frac{\lambda_1 e^{v_n}}{\int_{\Omega} e^{v_n} \,dx} - \frac{\lambda_1 e^{-v_n}}{\int_{\Omega} e^{-v_n}\, dx}\quad \text{in}\quad \Omega \tag{3} \]
with the boundary condition \(v_n = 0\) on \(\partial \Omega\) for a smooth bounded domain \(\Omega\) in \({\mathbb R}^2\). The functional associated to (3) is
\[ J_{\lambda_1, \lambda_2} (u) = \frac{1}{2}\int_{\Omega} | \nabla u| ^2 \,dx - \lambda_1 \log\left(\int_\Omega e^u \,dx\right)-\lambda_2 \log\left(\int_\Omega e^{-v} \,dx\right). \]
It is known in [H. Ohtsuka and T. Suzuki, Adv. Differ. Equ. 11, 281–304 (2006; Zbl 1109.26014) and I. Shafrir and G. Wolansky, J. Eur. Math. Soc. 7, 413–448 (2005; Zbl 1129.26304)] that if \(\lambda_1 \leq 8\pi\) and \(\lambda_2 \leq 8\pi\), then there exists a positive constant \(C\) such that
\[ J_{\lambda_1, \lambda_2} (u) \geq -C\quad \text{for}\quad u\in H^1_0(\Omega). \]
The authors prove that if \(\Omega\) is a non-simply connected domain in \({\mathbb R}^2\) and \(\lambda_1 \in (8\pi, 16\pi)\) and \(\lambda_2 \in (0, 8\pi)\), then the equation (3) has a solution.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
58J05 Elliptic equations on manifolds, general theory
35B40 Asymptotic behavior of solutions to PDEs
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E20 Harmonic maps, etc.
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