Existence of solutions for Kirchhoff-type fractional Dirichlet problem with $p$-Laplacian

TL;DR

This paper proves the existence of at least one non-trivial weak solution for a fractional Kirchhoff-type p-Laplacian system using mountain pass theorem, providing bounds on the parameter and solution behavior as the parameter grows.

Contribution

It establishes the existence of solutions for a fractional Kirchhoff p-Laplacian system with new bounds on the parameter and solution estimates, extending previous results to fractional derivatives.

Findings

Existence of at least one non-trivial weak solution for large parameter λ.

Derived concrete lower bounds for λ and estimates for solutions.

Showed that solutions tend to zero as λ approaches infinity.

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 $\lambda$. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution $u_\lambda$ under some local superquadratic conditions for each given large parameter $\lambda$. We get a concrete lower bound of the parameter $\lambda$, and then obtain two estimates of weak solutions $u_\lambda$. We also obtain that $u_\lambda\to 0$ if $\lambda$ tends to $\infty$. Finally, we present an example as an application of our results.