Multiplicity Of Solutions For Elliptic Equation

This work deals with the existence of multiplicity solutions for quasilinear Euler-Lagrange equa-tion. with zero Dirichlet boundary condition.The non-differentiable functional in relation with the above equation is defined providing that 1<θ<p<q<p*/p(γ+p),γ> 1,1<p≤<N and λ>0.By using critical points method we establish the existence of multiplicity solutions for the above equation in the following cases:If 1<θ<p<q<p*/p(γ+p) and there is a constant λ0>0 such that 0<λ<λ0, the equation has an infinitely many bounded weak solutions.If 1<θ<N<q<N+γ and there is a nonnegative constant λ* such that 0<λ<λ*, the equation possesses an infinitely many bounded weak solutions.If 1<θ<p<γ+p<q<p*/p(γ+p), and there is a nonnegative constant Λ such that λ<Λ, the equation has at least two positive bounded weak solutions.