Of Symmetric Dirichlet Problem For Quasilinear Elliptic Equations Disturbance

In this paper, we study the multiplicity results of solutions for a symmetric quasilinear boundary value problem with perturbation by a nonsymmetric term, by using minimax methods from critical point theory.Consider the perturbation for quasilinear boundary value problemwhere Ω. (?) R^N is a bounded domain with smooth boundary is the p-Laplacian operator, e is a parameter, g : Ω×R â†'R is an arbitrary continuous function, and f : Ω×R â†'Risa continuous function. We impose the function f two classes of conditions as follows.The first class of conditions are global conditions:(f₁) f(x,-t) = -f(x, t), for all x ∈ Ω, t∈ R.(f₂) For 1 < p < N, there exist C > 0 and 1 < q < p^* — 1, where p^* =Np/(N-p), such that|f(x, t)| ≤ C(1 + |t|^q), for all x ∈Ω, t ∈ R,and for N = p ,(f₃) There exist M > 0 and μ > p , such that0 < μF(x, t) ≤ tf{x, t), for all x∈Ω, |t| ≥ M,where F(x,t) = f₀^t f(x,s) ds.