摘要：We investigate nonexistence results of nontrivial solutions of fractional differential inequalities of the form ( FS q m ) : { D 0 / t q x i − Δ H ( λ i x i ) ≥ η α i + 1 x i + 1 β i + 1 , ( η , t ) ∈ H N × ] 0 , + ∞ [ , 1 ≤ i ≤ m , x m + 1 = x 1 , $$\bigl(\mathrm{FS}^{m}_{q}\bigr)\mbox{:}\quad \left \{ \textstyle\begin{array}{l} \mathbf{D}^{q}_{0/t}x_{i}-\Delta_{\mathbb{H}}(\lambda_{i}x_{i}) \geq { \eta }^{\alpha_{i+1}} { x_{i+1} }^{\beta_{i+1}}, \quad (\eta ,t) \in{\mathbb{H}}^{N}\times\, ]0,+\infty [ , 1 \leq i \leq m, \\ x_{m+1}=x_) , \end{array}\displaystyle \right . $$ where D 0 / t q $\mathbf{D}^{q}_{0/t}$ is the time-fractional derivative of order q ∈ ( 1 , 2 ) $q \in(1,2)$ in the sense of Caputo, Δ H $\Delta_{\mathbb{H}}$ is the Laplacian in the ( 2 N + 1 ) $(2N+1)$ -dimensional Heisenberg group H N ${\mathbb {H}}^{N}$ , η ${ \eta }$ is the distance from η in H N ${\mathbb {H}}^{N}$ to the origin, m ≥ 2 $m\geq2$ , α m + 1 = α 1 $\alpha_{m+1}=\alpha_)$ , β m + 1 = β 1 $\beta _{m+1}=\beta_)$ , and λ i ∈ L ∞ ( H N × ] 0 , + ∞ [ ) $\lambda_{i}\in L^{\infty}({\mathbb{H}}^{N} \times\, ]0,+\infty [ )$ , 1 ≤ i ≤ m $1 \leq i \leq m$ . The main results are concerned with Q ≡ 2 N + 2 $Q \equiv2N + 2$ , less than the critical exponents that depend on q, α i $\alpha_{i}$ , and β i $\beta_{i}$ , 1 ≤ i ≤ m $1 \leq i \leq m$ . For q = 2 $q=2$ , we deduce the results given by El Hamidi and Kirane (Abstr. Appl. Anal. 2004(2):155-164, 2004) and El Hamidi and Obeid (J. Math. Anal. Appl. 208(1):77-90, 2003) from the hyperbolic systems. For m = 1 $m=1$ , we study the scalar case ( FI q ) : D 0 / t q x − Δ H ( λ x ) ≥ η α x β , $$(\mathrm{FI}_{q})\mbox{:}\quad \mathbf{D}^{q}_{0/t}x - \Delta_{\mathbb{H}}(\lambda x) \geq { \eta }^{\alpha} { x }^{\beta}, $$ where β > 1 $\beta>1$ , α are real parameters. In the last case, for q = 2 $q=2$ , we return to the approach of Pohozaev and Véron (Manuscr. Math. 102:85-99, 2000) from the hyperbolic inequalities.
关键词：critical exponent ; fractional derivative ; Heisenberg group ; evolution inequalities ; test function method
