Kirchhoff-type equations involving the Fractional $(p,q)-$Laplacian

TL;DR

This paper investigates the existence and nonexistence of solutions for a Kirchhoff-type fractional $(p,q)$-Laplacian problem, employing variational methods and critical point theory to establish multiple solutions under certain conditions.

Contribution

It introduces new existence results for solutions to a fractional Kirchhoff problem involving the $(p,q)$-Laplacian, using variational techniques and establishing nonexistence for small parameters.

Findings

Existence of at least two weak solutions via variational methods.

Application of Mountain Pass Theorem to obtain a second solution.

Nonexistence of solutions for small positive parameter values.

Abstract

In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional $(p\text{-}q)$-Laplacian problem: \begin{equation*} \begin{cases} M\left([u]^p_{p,s_1}\right)(-\Delta)^{s_1}_p u + M\left([u]^q_{q,s_2}\right)(-\Delta)^{s_2}_q u = \lambda\big[a(x)|u|^{p-2}u + b(x)|u|^{q-2}u\big] + h(x), & \text{in } \Omega, \\ u = 0, & \text{on } \mathbb{R}^N \setminus \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$ ($N \geq 1$) is a bounded domain with smooth boundary, $0 < s_1 < s_2 < 1$, and $s_1 p < N$. We assume $1 < q \leq p < \theta p < p^{*}_{s_1} := \dfrac{Np}{N - s_1 p}$, and $\lambda \in \mathbb{R}$. The functions $a(x), b(x)$, and $h(x)$ are non-negative, with $a, b \in L^\infty(\Omega)$ and $h \in L^q(\Omega)$. Using variational methods, we establish the existence of at least two weak solutions. The first solution is obtained via the direct minimization of the associated energy functional, and the second is obtained by applying the Mountain Pass Theorem. We also prove a nonexistence result for small values of the parameter $\lambda > 0$.