Transactions of the American Mathematical Society· 2014Q1
Existence of groundstates for a class of nonlinear Choquard equations
- 415citations
- Q1SCImago
- 2014year
Short summary
Researchers proved the existence of nontrivial groundstate solutions for a class of nonlinear Choquard equations using conditions inspired by Berestycki and Lions.
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Key points
- Proved existence of nontrivial groundstate solutions for a class of nonlinear Choquard equations.
- The proof uses conditions on the nonlinearity $F$ inspired by Berestycki and Lions.
- Solutions exhibit additional local regularity properties.
- Under specific conditions (even and monotone $F$), solutions are sign-constant and radially symmetric.
AI-generated from the title and abstract; the full text is not read.
Abstract
We prove the existence of a nontrivial solution $u \in H^1 (\mathbb {R}^N)$ to the nonlinear Choquard equation \[ - \Delta u + u = \bigl (I_\alpha \ast F (u)\bigr ) Fâ (u) \quad \text {in \(\mathbb {R}^N\),} \] where $I_\alpha$ is a Riesz potential, under almost necessary conditions on the nonlinearity $F$ in the spirit of Berestycki and Lions. This solution is a groundstate and has additional local regularity properties; if moreover $F$ is even and monotone on $(0,\infty )$, then $u$ is of constant sign and radially symmetric.
The authors' abstract, as published at the source. Transactions of the American Mathematical Society, 2014 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics