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

Abstract

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.

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

“…Remark From Li et al, the following relationship holds, for any , 1 < ϱ < ∞ . Nextly, the definition of weak solution for the BVP can be given based upon Remark .…”

“…Lemma 2.5. According to Li et al, 24 it is well known that the space E is a reflexive and separable Banach space. Lemma 2.6.…”

Multiplicity Results to a Conformable Fractional Differential Equations Involving Integral Boundary Condition

“…It is easy to obtain that , are the lower and upper solutions of initial value problem (34). By the help of Theorem 10, we deduce that problem (34) has existence solution ( ) with ( ) ≤ ( ) ≤ ( ).…”

“…Now there are more and more articles to prove that there are multiple solutions for integral boundary [29][30][31][32][33][34][35]. For example, [31] introduced the system of fractional differential equations…”

Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance

“…where C D α 1-and D become a popular research field. At present, many researchers study the existence of solutions of fractional differential equations such as the Riemann-Liouville fractional derivative problem at nonresonance [6][7][8][9][10][11][12][13][14][15][16], the Riemann-Liouville fractional derivative problem at resonance [17][18][19][20][21][22][23], the Caputo fractional boundary value problem [6,24,25], the Hadamard fractional boundary value problem [26][27][28], conformable fractional boundary value problems [29][30][31][32], impulsive problems [33][34][35], boundary value problems [8,[36][37][38][39][40][41][42][43], and variational structure problems [44,45].…”