Abstract
In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters.
Highlights
We study the existence of solutions for the following Choquard–Kirchhoff type equations
We point out that very recently in [47], Biswas et al firstly proved a embedding theorem for variable exponential Sobolev spaces and Hardy–Littlewood–Sobolev type result, and they studied the existence of solutions for Choquard equations as follows
Motivated by the above cited works, we find that there are no results for Choquard–Kirchhoff type equations involving a variable s( x, ·)-order fractional p1 ( x, ·)&p2 ( x, ·)Laplacian
Summary
We study the existence of solutions for the following Choquard–Kirchhoff type equations. For Choquard–Kirchhoff equations with variable exponent in [18], Bahrouni et al dealt with Strauss and Lions type theorems and studied the existence and multiplicity of weak solutions. Studied the multiplicity results for a Schrödinger equation via variational methods Most importantly, they obtained the embedding theorem for variable-order Sobolev spaces. We point out that very recently in [47], Biswas et al firstly proved a embedding theorem for variable exponential Sobolev spaces and Hardy–Littlewood–Sobolev type result, and they studied the existence of solutions for Choquard equations as follows (−∆) p(x,·) v( x ). In the whole space R N , a new variable-order fractional p( x, ·)-Kirchhoff type problem under two kinds of weaker conditions was studied in [51].
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