Non-homogeneous $(p_1,p_2)$-fractional Laplacian systems with lack of compactness

TL;DR

This paper investigates the existence of weak solutions for non-homogeneous fractional Laplacian systems with variable exponents, using variational methods and Nehari manifold techniques under certain boundedness conditions.

Contribution

It introduces a novel approach to establish weak solutions for non-homogeneous fractional systems with variable exponents via variational methods and dual norm bounds.

Findings

Existence of weak solutions under specific conditions.

Application of Nehari manifold minimization techniques.

Handling of non-homogeneous terms with bounded dual norms.

Abstract

The present paper studies the existence of weak solutions for the following type of non-homogeneous system of equations \begin{equation*} (S) \left\{\begin{aligned} (-\Delta)^{s_1}_{p_1} u &=u|u|^{\alpha-1}|v|^{\beta+1}+f_1(x) \,\mbox{ in }\, \Omega, \\ (-\Delta)^{s_2}_{p_2} v &=|u|^{\alpha+1}v|v|^{\beta-1}+f_2(x) \,\mbox{ in }\, \Omega, \\ u=v &= 0 \,\mbox{ in }\, \mathbb{R}^N \setminus \Omega, \\ \end{aligned} \right. \end{equation*} where $\Omega \subset \mathbb{R}^N$ is smooth bounded domain, $s_1,s_2 \in (0,1)$, $1<p_1,p_2<\infty$, $N>\max\{p_1s_1,p_2s_2\}$, $\alpha>-1$ and $\beta>-1$. We employ the variational techniques where the associated energy functional is minimized over Nehari manifold set while imposing appropriate bound on dual norms of $f_1,f_2$.