Abstract
In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p · -fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.
Highlights
The fractional differential operators and equations have increasingly attracted much attention, since they are good at describing memory and heredity of some complex systems compared with the integer-order derivative [1, 2]
The fractional differential operators have been applied in various research fields, such as optimization [3], fractional quantum mechanics [4], finance [5], image process [6], and biomedical engineering [7]
The variable-order fractional derivative extends the study of constant order fractional derivative, which was first proposed by Samko and Ross [11] in 1993
Summary
The fractional differential operators and equations have increasingly attracted much attention, since they are good at describing memory and heredity of some complex systems compared with the integer-order derivative [1, 2]. The extensive applications urgently need systematic studies on the existence, uniqueness of solutions to these variable-order fractional differential equations. In [18], the infinitely many solutions to Kirchhoff-type variable-order fractional Laplacian equations have been discussed. Considering that for some nonhomogeneous materials, the commonly used methods in Lebesgue and Sobolev spaces LpðΩÞ and W1,pðΩÞ are not sufficient; many scholars have begun to study the differential operator with variable exponent [21,22,23]. Pucci et al [34] studied a Kirchhoff-type eigenvalue problem which has a critical nonlinearity and nonlocal fractional Laplace. The results of the variable-order fractional Sobolev spaces with variable exponents and fractional pð·Þ-Laplace equations with variable order are few.
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