Existence of solutions for the fractional Kirchhoff equations with sign-changing potential

Abstract

In this paper, the authors investigate the following fractional Kirchhoff boundary value problem:where the parameter λ > 0 and constants a, b > 0. By applying the mountain pass theorem and the linking theorem, some existence results on the above fractional boundary value problem are obtained. It should be pointed out that the potential V may be sign-changing.

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:…”

“…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 [18‐23], 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: $$$$ \left\{\begin{array}{l}{{\left(a+b{\int}_0^T{\left|{}_0{D}_t^{\alpha }u(t)\right|}^p dt\right)}^{p-1}}_t{D}_T^{\alpha }{\phi}_p\left({}_0{D}_t^{\alpha }u(t)\right)=f\left(t,u(t)\right),\kern0.30em t\in \left(0,T\right),\\ {}u(0)=u(T)=0,\end{array}\right.$$…”

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

“…It is obvious that system (1.1) is much more complicated than system (1.4) since the appearance of nonlocal term A(u(t)) and p-Laplacian term φ p (s). Recently, in [30], the following fractional Kirchhoff equation with Dirichlet boundary condition was investigated…”

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

“…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.…”