Existence and multiplicity of positive solutions for a class of nonlinear elliptic problems

We study the existence and multiplicity of nonnegative solutions for the nonlinear elliptic problem, -Delta u+v(x)u=a(x)u^p+lambda f(x,u) for xinOmega and u=0 on partialOmega, where Omega is a bounded region in R^N, N 2, 10 and f(x,u) satisfies some suitable conditions. By extracting the Palais-Smale sequences in the Nehari manifold, it is proved that there exists lambda^* such that for lambda in (0,lambda^*), the above problem has at least two positive solutions.