Positive solutions for quasilinear system by using of weak sub-super solutions method and energy function

‎In this paper we consider the quasilinear system‎‎\[‎‎\left\{\begin{array}{ll}‎-‎\Delta_{p}u=\lambda g(x)f_{۱}(u,v)+\mu h_{۱}(u)‎ ~~~~ ‎in~‎~ ‎x\in\Omega‎‎\\\\‎-‎\Delta_{q}v=\lambda g(x)f_{۲}(u,v)+\mu h_{۲}(v)‎ ~~~~ ‎in~‎~ ‎x\in \Omega‎‎\\\\‎‎u(x)=v(x)=۰‎ ~~~~ ‎on‎ ~~ ‎x\in\partial\Omega~~‎.‎\end{array}\right‎.‎\]‎‎\\‎‎where‎, ‎\Omega is a bounded domain in \mathbb{R}^n with smooth‎‎boundary \partial\Omega, \lambda>۰ is a parameter‎. ‎Here g is‎‎C^۱ sign-changing function that may be negative near the boundary‎‎and h_{۱},h_{۲},f_{۱},f_{۲} are nondecreasing and satisfy in‎‎additional conditions that we shall express the follow‎. ‎Also we‎‎introduce the energy functional associate our problem then by energy‎‎functional we will discuss in context existence of solution for‎‎said problem‎.‎\end{abstract}‎

کلیدواژه ها:

Positive solutions ، quasilinear‎ ‎elliptic system ، supersolution and subsolution ، energy functional‎