positive solutions for quasilinear system by using of weak sub-super solutions method and energy functional

‎in this paper we consider the quasilinear system‎‎[‎‎left{begin{array}{ll}‎-‎delta_{p}u=lambda g(x)f_{1}(u,v)+mu h_{1}(u)‎ ~~~~ ‎in~‎~ ‎xinomega‎‎‎-‎delta_{q}v=lambda g(x)f_{2}(u,v)+mu h_{2}(v)‎ ~~~~ ‎in~‎~ ‎xin omega‎‎‎‎u(x)=v(x)=0‎ ~~~~ ‎on‎ ~~ ‎xinpartialomega~~‎.‎end{array}right‎.‎]‎‎‎‎where‎, ‎omega is a bounded domain in mathbb{r}^n with smooth‎‎boundary partialomega,lambda>0 is a parameter‎. ‎here g is‎‎c^1 sign-changing function that may be negative near the boundary‎‎and h_{1},h_{2},f_{1},f_{2} 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.