Extistence Of Solutions For P(x)-Laplacian Equation With Cerami Condition

Under the scope of the variational method,the paper study the existence of fhe solution of the problem(p).Generally,we call partial differential equations similar to the form following the p(x)一Laplacian Dirichlet problem.Ω(?)Rn,Ω is a bounded domain, f:Ω×R�'R,(ρ)isThe problem is called as the p(x)-Laplacian Dirichlet problem.Since the birth of partial differential equations has been the mathematical study of hot pot.The predecessors mathematician Ambrosetti-Rabinowitz proposed(AR)conditions when they studied elliptic equations. The condition becomes an important condition of variational method.XLFan and other mathematician spread(AR)conditionsto the following form:(f1),∈C(Ω×R)and satisfies the Carathedory condition and α∈C(Ω),p(x)<α(x)<p*(x)=np(x)/n-p(x)(?)c,cis a constant s.t f(x,u)≤c(1+|u|α(x)-1)；(f2):(?)θ>p+,|t|≥M,x∈Ω,st. θ<θF(x,u)≤uf(x,u) whereThe conditions (f2) is the famous(AR)condition.The condition plays an important role in proving functionals satisfy the(PS)condition.The paper introduces the the Cerami conditions replace(PS)condition,and introduces a new deformation lemma instead of the Mountain Pas s Lemma.(Deformation Lemma with Cerami Condition):X is a B anach space,extst φ∈(X,R),e∈X,r>0,s.t If φ satisfies Cerami condition,and where Γ:={y∈C([0,1],X):y(0)=0,y(1)=e} then c is the critical value of φ.The conditions that we need to replace(AR)condition are listed as follow:(H1) f:Ω×R�'R and f satisfies Carathedory condition,and α∈C(Ω),p(x)<α(x)<p*(x),|f(u)|≤C(1+|t|α(x)-1),(?)(x,t)∈Ω×R hold: uniformly for a.e.z∈Ω；(H3)(?)θ≥1,such that for any t∈Rand5∈[0,1]and each μ∈[p-,p+],and all of Gλ∈f the inequalities θGλ(x,t)≥Gu(x,st)hold for a.e.x∈Ω, where(?)={Gλ:Gλ(x,t)=f(x,t)t-λF(x,t),λ∈[p-,p+]}；(H4)f(x,t)=o(|t|p+-1),as t�'0uniformly for a.e.x∈Ω；(H5)There are potives N,C1,C2and a>p such that: C1|t|p(x)[ln(e+|t|)]a(x)-1≤C2tf(x,t)/ln(e+|t|)≤tf(x,t)一p(x)F(x,t),(?)|t|≥N,(?)x∈Ω；(H6) There is a vector1,1∈Rn\{0}, and Vx∈Ω,ρ(t)=p(x+tl) is monotone,for t∈F.(t)={t|x+1t∈Ω).The main conclusions of this paper are as follow:Theorem3.1If f satisfies conditions(H1),(H2),(Ha),(H4)then the problem(p) has non-trivial solution.Theorem3.2If f satisfies conditions(H1),(H5),(H4),(H6)then there is at least a non-trivial solution of(ρ).