The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate

Abstract

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$-\Delta u+au=\frac{1}{2\pi }[\ln \frac{1}{|x|}\ast |u{|}^2]u\text{in R2}$$ 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).

arXiv comment21 pages

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• Existence and multiplicity of solutions for the fractional $p$-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth
• Nonlocal planar Schrödinger-Poisson systems in the fractional Sobolev limiting case
• Elliptic problem driven by different types of nonlinearities
• Schrödinger-Poisson systems with zero mass in the Sobolev limiting case
• The Choquard logarithmic equation involving a nonlinearity with exponential growth
• Existence and axial symmetry of minimal action odd solutions for 2-D Schrödinger-Newton equation
• On the existence and multiplicity of solutions for the $N$-Choquard logarithmic equation with exponential critical growth