Abstract
In this paper, we consider the fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity. By means of the concentration-compactness principle in fractional Sobolev space and the Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions, which tend to zero for suitable positive parameters.
Highlights
The main purpose of this paper is to study the existence and multiplicity of solutions for the p-fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity
Nonlocal operators can be seen as the infinitesimal generators of Lévy stable diffusion processes [2]
They allow us to develop a generalization of quantum mechanics and to describe the motion of a chain or an array of particles that are connected by elastic springs as well as unusual diffusion processes in turbulent fluid motions and material transports in fractured media
Summary
Authors investigated the existence of solutions for Kirchhoff-type problem involving the fractional p-Laplacian via variational methods, where the nonlinearity is subcritical, and the Kirchhoff function is nondegenerate. In [31], the authors studied a nonlocal equation involving the fractional p-Laplacian (−∆)spu + V (x)|u|p−2u = f (x, u) + λh in Rn. When the nonlinearity f is assumed to have exponential growth, by using a fixed point method, the authors established an existence result on weak solutions. By using the mountain pass theorem and Ekeland’s variational principle, the authors in [40] studied the multiplicity of solutions to a nonhomogeneous Kirchhoff-type problem driven by the fractional p-Laplacian, where the nonlinearity is convex-concave, and the Kirchhoff function is degenerate.
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