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Nondegeneracy of entire solutions of a singular Liouvillle equation. (English) Zbl 1242.35117

This article deals with the model problem \[ \Delta u +e^u=4\pi N\delta_0 \quad \text{in } \,\, \mathbb R^2 , \] where \(\delta_0\) is the Dirac mass at the origin and \(N\) is a nonnegative integer.
It is known that all solutions of this problem with finite mass \(\int_{\mathbb R^2}e^u<+\infty\) are given by the family \[ U_{\mu,a}=\log \frac{8\mu^2(N+1)^2|z|^{2N}}{(\mu^2+|z^{N+1}-a|^2)^2},\,\, \mu\in \mathbb R\,\, a=a_1+ia_2\in C. \] The authors prove, that if \(\phi\in L^2(\mathbb R^2)\) solves the linearized equation \[ L(\phi):=\Delta \phi+e^{U_{\mu,a}}=0, \] then \(\phi\) must be a linear combination of the functions \(Z_1:=\partial_{\mu}U_{\mu,a}\), \(Z_2:=\partial_{a_1}U_{\mu,a}\), \(Z_3:=\partial_{a_2}U_{\mu,a}\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
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