2020/03/26 by Ahmad Z. Fino, Fino, Ahmad Z., Mohamed Jleli +3
Mathematics · #35B33 #35B53 #35R11 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2003.12167
openalex publication_date 2020/03/26 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We consider the fractional elliptic inequality with variable-exponent\nnonlinearity (-
Delta)^
frac
alpha2 u+
lambda
,
Delta u
geq\n|u|p(x),
quad x
in
mathbbRN, where N\≥ 1, \α\∈ (0,2),\n\λ\∈\ℝ is a constant, p: \ℝN\→ (1,\∞) is a\nmeasurable function, and (-\Δ)\(\α)/(2) is the fractional\nLaplacian operator of order \(\α)/(2). A Liouville-type theorem is\nestablished for the considered problem. Namely, we obtain sufficient conditions\nunder which the only weak solution is the trivial one. Next, we extend our\nstudy to systems of fractional elliptic inequalities with variable-exponent\nnonlinearities. Besides the consideration of variable-exponent nonlinearities,\nthe novelty of this work consists in investigating sign-changing solutions to\nthe considered problems. Namely, to the best of our knowledge, only\nnonexistence results of positive solutions to fractional elliptic problems were\ninvetigated previously. Our approach is based on the nonlinear capacity method\ncombined with a pointwise estimate of the fractional Laplacian of some test\nfunctions, which was derived by Fujiwara (2018) (see also Dao and Reissig\n(2019)). Note that the standard nonlinear capacity method cannot be applied to\nthe considered problems due to the change of sign of solutions.\n