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Positive solutions for multi-parameter cyclic $(p_1,\dots,p_n)$-Laplacian systems with combined and falling-zero nonlinearities | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 04 اسفند 1404 اصل مقاله (435.66 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2025.39875.5586 | ||
| نویسنده | ||
| Mahdi Choubin* | ||
| Department of Mathematics, Razi University, Kermanshah, Iran | ||
| تاریخ دریافت: 07 آذر 1404، تاریخ بازنگری: 02 دی 1404، تاریخ پذیرش: 10 دی 1404 | ||
| چکیده | ||
| We consider a cyclic $n\times n$ system of quasilinear elliptic equations \[ \begin{cases} - \Delta_{p_i} u_i = \lambda_i f_i(u_i) + \mu_i g_i(u_{i+1}), & i=1,2,\dots,n-1,\\%[1mm] - \Delta_{p_n} u_n = \lambda_n f_n(u_n) + \mu_n g_n(u_1), \end{cases} \] in a bounded smooth domain $\Omega\subset\mathbb{R}^N$ with homogeneous Dirichlet boundary conditions, where $\Delta_{p_i}z=\operatorname{div}(|\nabla z|^{p_i-2}\nabla z)$, $p_i>1$, and $\lambda_i,\mu_i>0$. The nonlinearities $f_i,g_i:[0,\infty)\to\mathbb{R}$ are increasing and satisfy a subcritical growth condition at infinity for $f_i$ and a combined sublinear composition condition for the cooperative chain $(g_i)_{i=1}^n$. This setting includes power-type and piecewise power-type nonlinearities. Under these assumptions, we prove the existence of positive weak solutions for all sufficiently large values of the sums $\lambda_i+\mu_i$. Under additional flatness conditions near the origin we obtain at least two distinct positive solutions. We also treat the case where $f_i$ have a falling-zero structure ($f_i>0$ on $(0,r_i)$, $f_i(r_i)=0$, $f_i<0$ on $(r_i,\infty)$) and derive analogous existence and multiplicity results. The proofs rely on the method of sub- and supersolutions and a three-solution theorem in an ordered Banach space. | ||
| کلیدواژهها | ||
| Multiple parameters؛ $(p_1؛ p_2؛ \dots؛ p_n)$-Laplacian systems؛ Combined sublinear effects؛ Falling zeroes؛ Sub- and supersolutions | ||
| مراجع | ||
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آمار تعداد مشاهده مقاله: 27 تعداد دریافت فایل اصل مقاله: 58 |
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