The Existence Of Solution For Nonlinear Elliptic Differential Equations

This paper discussed the existence of solutions for the singular quasilinear elliptic problems-â–³_pu=f(x,u) x∈ΩwhereΩis a bounded in Rⁿ,n≥1,â–³_p(x)u:=div(|â–½u|^p(x)-2â–½u),â–³_p(x)is the p(x)-Laplacian, 1＜p＜∞, and f is a function onΩ×(0,∞).First, when p(x)=p is a constant, this paper discussed the existence of solutions fbr elliptic problems:where f(x, t) is a Caratheodory function onΩ×(0,∞) and satisfying some construct conditions. This paper improve the work of Ravi P.Agarwal and others in 2006 [1].Secondly, when p(x) is continuous function on (?), we discuss the following p(x)-lapalace problem with nontrivial solutions:where f(x, t) is a continuous onΩ×(0,∞) and satisfy some of the critical conditions for growth. The results improve part of the work of Fan Xianling in 2003 [3].