Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent
(2015) Communications in Contemporary Mathematics — Vol. 17, n° 05, p. 1550005 (2015)
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- Abstract
- We consider nonlinear Choquard equation −Δu + Vu = (Iα * |u|^(α/N + 1))|u|^(α/N - 1) u where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power in the nonlocal part of the equation is critical with respect to the Hardy–Littlewood–Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if liminf∞ (1-V(x))|x|²>N²(N - 2)/(4(N + 1)) then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis–Lieb lemma.
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Moroz, V., & Van Schaftingen, J. (2015). Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent. Communications in Contemporary Mathematics, 17(05), 1550005. https://doi.org/10.1142/S0219199715500054 (Original work published 2015)