The Existence Of Nontrivial Solutions For A Class Of P-Laplacian Equations

In this paper,by using the Nehari method and variational method,we study the existence of nontrive solutions for some p-Laplacian equations.Firstly,we consider the following p-Laplacian problem with subcritical ex-ponent(?)where ?(?)RN(N? 3)is an open bounded domain with smooth boundary,p<q<r<p*,p*=Np/N-p is the critical Sobolev exponent.We obtain the existence of nontrivial solution for problem(0.1)via the Nehari method and vari-ational methods.Theorem 0.1.If V ? 0,equation(0.1)has a nontrivial solution.Next,we consider the following p-Laplacian equations where 1<p<N.We assume that the potential V verifies the following hy-potheses:(V1)There exists ?>0,such that V(x)??>0,x ? RN.(V2)V(x)= V(x + y),x?RN,y?ZN.The function f? C(R,R)is written as K(s)=f0(s)+?g(s)Where e is a real parameter.f0,g are locally Holder continuous functions satis-fying the conditions:(F1)f0(0)=g(0)and g(s)?0 for all s?0.(F2)f0(t)=o(|t|)as |t|?0,and g(t)=o(|t|)as |t|?0.(F3)There exists q?(2p,2p*)such that |f0(s)|?|s|q-1,for all s?R.(F5)There exists a sequence of positive real numbers {Mn},where {M}?+?,such that(F6)For ?>0 given by(V1),there exists l>p and T?(0,(l/p-1)?)such that We obtained the following theorem:Theorem 0.2.Assume that(V1)-(V2)and(F1)-(F6)hold,then there is a?0>0 such that(0.2)has a nontrivial solution for all 0<? ??0.