The existence of solutions for an impulsive fractional coupled system of (p, q)‐Laplacian type without the Ambrosetti‐Rabinowitz condition

Abstract

In this article, based on the variational approach, the existence of at least one nontrivial solution is studied for (p, q)‐Laplacian type impulsive fractional differential equations involving Riemann‐Liouville derivatives. Without the usual Ambrosetti‐Rabinowitz condition, the nonlinearity f in the paper is considered under some suitable assumptions.

Existence and multiplicity of solutions for (p,q)$$ \left(p,q\right) $$‐Laplacian Kirchhoff‐type fractional differential equations with impulses

“…for any t ∈ R, 0 < x < 𝑦 or 𝑦 < x < 0, where  (t, x) = x𝑓 (t, x) − qF(t, x), F(t, x) = ∫ x 0 𝑓 (t, s)ds. Significantly, Li et al [9] expanded the (MS)-condition for nonlinearity 𝑓 (t, x, 𝑦), that is, there exists a constants G * > 0 such that (t, 𝜁 x, 𝜁𝑦) ≤ (t, x, 𝑦) + G * , (1.4) for any x, 𝑦 ∈ R, 𝜆 > 0 and 0 < 𝜁 ≤ 1, where (t, x, 𝑦) = 𝑓 x (t, x, 𝑦)x + 𝑓 𝑦 (t, x, 𝑦)𝑦−𝜆𝑓 (t, x, 𝑦). In general, the (MS)-condition is crucial in demonstrating that the Palais-Smale sequence or Cerami-sequence is bounded when the nonlinearities that are superlinear do not satisfy the (AR)-condition.…”

“…In recent years, a weaker condition than the (AR)-condition, firstly introduced by Miyagaki and Souto ((MS)-condition [8] for short), has been widely used in applying the critical point theory [9][10][11][12][13].…”

“…(MS)‐condition: There exist positive constants $$$ {\overline{C}}_{\ast },\overline{q}&gt;0 $$$ such that $$$$ \mathcal{F}\left(t,x\right)\le \mathcal{F}\left(t,y\right)&#x0002B;{\overline{C}}_{\ast }, $$$$ for any $$$ t\in R,0&lt;x&lt;y $$$ or $$$ y&lt;x&lt;0 $$$, where $$$ \mathcal{F}\left(t,x\right)&#x0003D; xf\left(t,x\right)-\overline{q}F\left(t,x\right),F\left(t,x\right)&#x0003D;{\int}_0&#x0005E;xf\left(t,s\right) ds $$$. Significantly, Li et al [9] expanded the (MS)‐condition for nonlinearity $$$ f\left(t,x,y\right) $$$, that is, there exists a constants $$$ {G}&#x0005E;{\ast }&gt;0 $$$ such that $$$$ \mathcal{H}\left(t,\overline{\zeta}x,\overline{\zeta}y\right)\le \mathcal{H}\left(t,x,y\right)...$$…”

Variational approach to instantaneous and noninstantaneous impulsive system of differential equations

“…They obtained that the problem (1.3) has at least one nontrivial solution via variational methods. For the recent works about instantaneous impulsive differential systems, the interested readers may refer to [31][32][33][34].…”

Variational formulation for nonlinear impulsive fractional differential equations with (p, q)‐Laplacian operator

“…Meanwhile, in recent years, owing to the appearance of the coexistence problem of p and q ‐Laplacian operators, equations including the ( p , q )‐Laplacian operator have been investigated 10–13 . For example, in Li et al, 14 by means of the mountain pass theorem, the existence of at least one nontrivial solution was discussed for an impulsive fractional coupled system with the generalized ( p , q )‐Laplacian operator …”