Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian

Abstract

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann-Liouville fractional derivatives and a parameter λ. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution u λ under some local superquadratic conditions for each given large parameter λ. We get a concrete lower bound of the parameter λ, and then obtain two estimates of weak solutions u λ . We also obtain that u λ → 0 if λ tends to ∞. Finally, we present an example as an application of our results. φ p (s) := |s| p−2 s, ∇F (t, x) is the gradient of F with respect to x = (x 1 , · · · , x N ) ∈ R N , that is, ∇F (t, x) = ( ∂F ∂x1 , · · · , ∂F ∂xN ) τ , and F : [0, T ] × R N → R satisfies the following condition:(H0) there exists a constant δ > 0 such that F (t, x) is continuously differentiable in x ∈ R N with |x| ≤ δ for a.e.t ∈ [0, T ], measurable in t for every x ∈ R N with |x| ≤ δ, and there exist a ∈ C(R + ,for all x ∈ R N with |x| ≤ δ and a.e. t ∈ [0, T ].When α = 1, the operator t D α T ( 0 D α t u(t)) reduces to the usual second order differential operator −d 2 /dt 2 . Hence, if α = 1, p = 2, N = 1, λ = 1 and V (t) = 0 for a.e. t ∈ [0, T ], system (1.1) becomes the equation with

Multiple Solutions for a Kirchhoff-type Fractional Coupled Problem with p-Laplacian

“…The variational structure of equation ( 1) becomes more complex and difficult to investigate as a result of these factors. As a result, our work complements and improves Chen et al [12] and Kang et al [16]. Specifically, when â 1) is equivalent to the following equation…”

“…Unlike the two types of fractional order differential equations listed above, the Kirchhoff-type fractional order differential equations discussed in this study have received less attention to my knowledge. (see [12][13][14][15][16] and references therein). In these articles, just certain asymptotic conditions for the non-linear terms on the right-hand side of the equation are given to derive the existence and multiplicity results for the equation's solutions, ignoring the geometric conditions for the non-linear terms, which are addressed in this paper.…”

“…In Section2, some important definitions and lemmas are given in this section. In Section 3, we conclude that there are at least two different weak solutions to the equation (1) by combining Lemma 5 and Lemma 6, rather than using the traditional mountain pass theorem as in [2,10,12,13,16], and then Theorem 1 is used to show that the problem has at least two non-trivial solutions based on geometric considerations. Finally, we bring the paper to a close in Section 4.…”

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

“…) is said to be non-degenerate Kirchhoff-type fractional differential equation. Thus, this paper broadens the findings of Chai and Liu [18], Chen et al [20], and Kang et al [22]. To my knowledge, some scholars have studied a class of Kirchhoff-type differential equations of integer order with impulse effects (see Caristi et al [29] and Afrouzi et al [30]), but this class of equations involving fractional order by variational methods is still an open question.…”

“…Some worthwhile studies of Kirchhoff equations include those by Graef et al [15], Hssini [16], Matallah et al [17], and references therein. It should be mentioned that the existence of solutions to the Kirchhoff-type fractional differential equations addressed by variational methods has also received much attention from scholars [18][19][20][21][22][23], as models based on fractional order are better suited to describing the memory and hereditary properties of many processes and materials. Here, by using the mountain pass theorem and genus properties in the critical point theory, Chen et al [20] were concerned with the following Kirchhoff-type fractional Dirichlet problem with p-Laplacian:…”