Existence of solutions for non-local elliptic systems with Hardy-Littlewood-Sobolev critical nonlinearities
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Date
2019-07-19
Authors
Yang, Yang
Hong, Qian Yu
Shang, Xudong
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this work, we establish the existence of solutions for the non-linear nonlocal system of equations involving the fractional Laplacian,
(-∆)su = αu + bv + 2p / p+q ∫Ω |v(y)|q / |x-y|μ dy|u|p-2u
+2ξ1 ∫Ω |u(y)2*μ / |x-y|μ dy|u|2*μ-2u in Ω,
(-∆)sv = bu + cv + 2q / p+q ∫Ω |u(y)|p / |x-y|μ dy|v|q-2vv
+2ξ2 ∫Ω |v(y)2*μ / |x-y|μ dy|v|2*μ-2v in Ω,
u = v = 0 in ℝN \ Ω,
where (-∆)s is the fractional Laplacian operator, Ω is a smooth bounded domain in ℝN, 0 < s < 1, N > 2s, 0 < μ < N, ξ1, ξ2 ≥ 0, 1 < p, q ≤ 2*μ and 2*μ = 2N-μ / N-2s is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality. The nonlinearities can interact with the spectrum of the fractional Laplacian. More specifically, the interval defined by the two eigenvalues of the real matrix from the linear part contains an eigenvalue of the spectrum of the fractional Laplacian. In this case, resonance phenomena can occur.
Description
Keywords
Fractional Laplacian, Choquard equation, Linking theorem, Hardy-Littlewood-Sobolev critical exponent, Mountain Pass theorem
Citation
Yang, Y., Hong, Q. Y., & Shang, X. (2019). Existence of solutions for non-local elliptic systems with Hardy-Littlewood-Sobolev critical nonlinearities. <i>Electronic Journal of Differential Equations, 2019</i>(90), pp. 1-32.
Rights
Attribution 4.0 International