Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 2026 02 23 Positive solutions for multi-parameter cyclic $(p_1,\dots,p_n)$-Laplacian systems with combined and falling-zero nonlinearities 10503 10.22075/ijnaa.2025.39875.5586 EN Mahdi Choubin Department of Mathematics, Razi University, Kermanshah, Iran Journal Article 2025 11 28 We consider a cyclic $n\times n$ system of quasilinear elliptic equations<br />\[<br />\begin{cases}<br />- \Delta_{p_i} u_i = \lambda_i f_i(u_i) + \mu_i g_i(u_{i+1}), & i=1,2,\dots,n-1,\\%[1mm]<br />- \Delta_{p_n} u_n = \lambda_n f_n(u_n) + \mu_n g_n(u_1),<br />\end{cases}<br />\]<br />in a bounded smooth domain $\Omega\subset\mathbb{R}^N$ with homogeneous Dirichlet boundary conditions,<br />where $\Delta_{p_i}z=\operatorname{div}(|\nabla z|^{p_i-2}\nabla z)$, $p_i>1$, and $\lambda_i,\mu_i>0$.<br />The nonlinearities $f_i,g_i:[0,\infty)\to\mathbb{R}$ are increasing and satisfy a subcritical growth<br />condition at infinity for $f_i$ and a combined sublinear composition condition for the cooperative chain<br />$(g_i)_{i=1}^n$. This setting includes power-type and piecewise power-type nonlinearities.<br />Under these assumptions, we prove the existence of positive weak solutions for all sufficiently large<br />values of the sums $\lambda_i+\mu_i$. Under additional flatness conditions near the origin we obtain at<br />least two distinct positive solutions. We also treat the case where $f_i$ have a falling-zero structure<br />($f_i>0$ on $(0,r_i)$, $f_i(r_i)=0$, $f_i<0$ on $(r_i,\infty)$) and derive analogous existence and<br />multiplicity results. The proofs rely on the method of sub- and supersolutions and a three-solution<br />theorem in an ordered Banach space. https://ijnaa.semnan.ac.ir/article_10503_186b690e29892f137b4c34cfa40a3a4d.pdf