The Nonlinear Boundary Value Problem Of A Class Of P-Laplace Equation Involving Critical Exponent

We are concerned with the existence of nontrival solutions to the nonlinear boundary value problem of quasilinear elliptic equation involving sobolev exponent as follows:whereΩis a bounded C¹ domain in Rⁿ ;Δ_p is for p-Laplace operator; 1 < p² < n,,α(x) is a nonnegative boundary measurable function inΩ;α(x) (?)0,α(x) attains its maximum at x₀ inΩand satisifies |α(x)—α(x₀)| =ο(|x- x₀|^p-1) as x→x₀; b(x) is a continuous function on (?)Ω, and b(x) > 0; (?) is the unit outer normal on dΩ.By the Mountain-Pass Theorem without (PS)condition,we find a sequence {u_m} which satisifies I{u_m)→C and I'(u_m)→0 as m→∞.we verify |â–½u_m|^p-2â–½u_m isweakly convergence by the concentration principal II only as At last,we verify this condition.So we solve our problem by five lemmas.