On the existence of three non-negative solutions for a $(p,q)$-Laplacian system

TL;DR

This paper proves the existence of at least three non-negative solutions for a coupled fractional $(p,q)$-Laplacian system using variational methods, under certain conditions on parameters and growth assumptions.

Contribution

It establishes the existence of multiple solutions for a nonlocal fractional system with general operators, extending previous results to a broader class of operators.

Findings

Existence of three non-negative solutions for the system.

Results hold under specific parameter ranges and growth conditions.

Extension to more general nonlocal operators $\\mathcal L_{\phi_i}$.

Abstract

The present paper studies the existence of weak solutions for \begin{equation*} (\mathcal{P}) \left\{\begin{aligned} (-\Delta)^{s_1}_{p_1} u &=\la f_1\,(x,u,v) +g_1(x,u) \,\mbox{ in }\, \Om, \\ (-\Delta)^{s_2}_{p_2} v &=\la f_2\,(x,u,v) +g_2(x,v) \,\mbox{ in }\, \Om, \\ u=v &= 0 \,\mbox{in }\, \Rn \setminus \Om, \\ \end{aligned} \right. \end{equation*} where $\Om \subset \Rn$ is a smooth bounded domain with smooth boundary, $s_1,s_2 \in (0,1)$, $1<p_i<\frac{N}{s_i}$, $i=1,2$, $f_i$ and $g_i$ has certain growth assumptions for $i=1,2$. We prove existence of at least three non negative solutions of $(\mathcal P)$ under restrictive range of $\lambda$ using variational methods. As a consequence, we also conclude that a similar result can be obtained when we consider a more general non local operator $\mathcal L_{\phi_i}$ instead of $(-\Delta)^{s_i}_{p_i}$ in $(\mathcal P)$.