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Construction of singular limits for a semilinear elliptic equation in dimension 2. (English) Zbl 0890.35047

The authors consider the Dirichlet problem for semilinear elliptic equations \[ \Delta u+\rho e^u = 0, \quad \text{in } \Omega, \qquad u=0, \quad \text{on } \partial \Omega,\tag{P} \] for a given regular bounded open subset \(\Omega\) of the complex plane \({\mathbb{C}}\). For any non-degenerate critical point \((z_1,\ldots,z_K)\) of a certain function \({\mathcal F}\), defined on \({\mathcal C}^K\) defined via the Green’s function of the domain \(\Omega\), the authors prove that there exists some \(\rho_0>0\) and a one parameter family of solutions \(\{u_{\rho}\}\) of the problem (P) with \(0<\rho\leq \rho_0\) such that the limit function \(u^{\ast}\) of \(u_{\rho}\) as \(\rho\to 0\) is a solution of the problem \[ -\Delta u^{\ast}= \sum_j8\pi \delta_{z_j}, \quad \text{in }\Omega, \qquad u^{\ast}= 0, \quad \text{on } \partial\Omega,\tag{P\(_0\)} \] with \((z_1,\ldots,z_K)\) as its singular points. This interesting result can be viewed as a partial solution of the converse problem of the earlier work of T. Sutuki on problem (P). Several works in a similar spirit have been done by V. H. Weston, J. L. Moseley, and other mathematicians in order to understand of the complete solution structure of related problems.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A35 Theoretical approximation in context of PDEs
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