Positive Solutions Of A Kind Of Equations Related To The Laplacian And P-Laplacian

Nonlinear partial differential equation usually arises in the natural science and engi-neering areas. Because they can well explain the important natural phenomenon, a large number of science researchers have paid attention to the problems for a long time. In this paper, we use the critical point theorem to study the existence of positive solution for a class of Laplace and p-Laplace equations. Our equation is the following form: where △pu=div(|(?)u|p-2(?)u),p>1 but p≠2,Ω is a smooth bounded domain in RN for N≥1, and f satisfies the following conditions: (f1)∈C(Ω×R,R),f(x,t)≥0 for any x∈Ω,t>0 and f(x,t)=0 for any x∈Ω,t≤0; (f2) For fo,f∞<∞, the limitsexist uniformly for x∈Ω. Let Our main result is the following theorem:Theorem Suppose that f satisfies (f1) and (f2), then(i) When 1<p< 2,f0<μ1,f∞> λ1. (P) has at least one positive solution;(ii) When p> 2,f0<λ1,f∞>μ1.(P) has at least one positive solution.