Solvability of fractional boundary value problem with p-Laplacian via critical point theory

Abstract

In this paper, we discuss the fractional boundary value problem containing left and right fractional derivative operators and p-Laplacian. By using critical point theory we obtain some results on the existence of weak solutions of such a fractional boundary value problem.MSC: 26A33; 34B15; 58E05

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

“…Motivated by the work mentioned above, in this paper, the first conclusion is given under satisfying A‐R condition for the fractional p ‐Laplacian differential systems in . We verify that BVP exists infinitely many solutions, which improves the results in Chen and Liu …”

“…As fractional differential BVPs have been coming into a meaningful research objective, a large number of researchers show enormous interest in it. Thus, many wonderful and important results have been presented, which have been widely applied in diverse fields in daily life, such as Zhao et al, Jiao and Zhou, Samko et al, Guo and Lakskmikantham, Miller and Ross, Hartley et al, Caputo, and Kilbas et al Especially, based on variational methods, most existence results of solutions have been established under the famous Ambrosetti‐Rabinowitz (A‐R) condition for fractional differential equation in recent years …”

“…By using critical point theorems, the existence results of at least 1 solution have been established for the fractional differential equation . Thereafter, Chen, in their study, investigated a p ‐Laplacian fractional differential equation as below: where and are the left and right Riemann‐Liouville derivatives with 0 < α ≤ 1, respectively. Functions Φ p ( s ) = | s | p −2 s , p > 1, and f ( t , u ( t )):(0, T )× R → R is continuous.…”

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

“…Thereafter, El‐Hamidi extended the (A‐R) condition for nonlinearity f ( t , x , y ), ie, there exist constants μ 1 > p , μ 2 > p and M 2 > 0 such that for any , , and can be as the consequence of , where are two positive constants. In general, the (A‐R) condition plays a necessary part to ensure the boundedness of Palais‐Smale sequences associated with the Euler‐Lagrange functional, and it arises in many relevant literatures frequently . For example, under assuming the (A‐R) condition, the authors concerned the existence of at least one solution for a fractional differential equation by using critical point theorems in Jiao and Zhou …”

“…On the other hand, because of the application of p ‐Laplacian operator (proposed by Leibenson) in non‐Newtonian fluid theory, nonlinear elastic mechanics, etc, the existence of solutions for p ‐Laplacian fractional BVPs was studied by using variational methods under the (A‐R) condition holds in resent years. By means of variational approaches, Chen and Liu considered the following p ‐Laplacian fractional equation: where 0 < α ≤ 1, and are the left and right Riemann‐Liouville derivatives, respectively. Φ p ( s ) = | s | p − 2 s , p > 1.…”