The Existence Of Nontrivial Solutions For P-Laplacian Equations Without(AR) Condition

In this paper,we study a class of nonlinear Dirichlet equation where N>p>1,△pu=div(|▽u|p-2▽u)is the p-Laplacian operator,Ω is a bounded smooth domain in RN(N ≥ 1),we assume without restriction that f(x,s)= 0,(?)s<0,a.e.x ∈ Ω we require that f(x,u)satisfies the following condition:(H1)f:Ω × R is a Caratheodory function,f(x,s)≥ 0,(?)s ≥ 0,x ∈ Ω;(H2)There is q ∈(p,Np/N-p)if N>p and N ≤ p if q>p,such that(?),uniformly for x ∈ Ω;(H3)(?)uniformly for x ∈ Ω;(H4)there is a ∈(0,∞],such that(?)uniformly for x ∈Ω.Under(H1)-(H4),we can obtain the existence of nontrivial solutions of this equation for parameter a ≤ +∞(condition(A1)and(A2)),referring to monotone method in[14],written by Jeanjean.