The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate
Copyright policy )The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate
We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - Δu + a u = \frac{1}{2 π} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).
21 pages
Positive ground state solution, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Semilinear elliptic equations, logarithmic Choquard equation, Positive solutions to PDEs, Planar Schrödinger–Newton system Positive ground state solution Asymptotics Nondegeneracy, Nondegeneracy, Mathematics - Analysis of PDEs, asymptotics, positive ground state solution, Planar Schrödinger–Newton system, FOS: Mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Analyse mathématique, Asymptotics, Analysis of PDEs (math.AP)
Positive ground state solution, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Semilinear elliptic equations, logarithmic Choquard equation, Positive solutions to PDEs, Planar Schrödinger–Newton system Positive ground state solution Asymptotics Nondegeneracy, Nondegeneracy, Mathematics - Analysis of PDEs, asymptotics, positive ground state solution, Planar Schrödinger–Newton system, FOS: Mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Analyse mathématique, Asymptotics, Analysis of PDEs (math.AP)
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