Multiple Solutions For Elliptic PDE Boundary Value Problems With Concave-convex Nonlinear Terms

In this paper,we mainly investigate two problems.One is the existence of solutions of semi-linear elliptic equation with Dirichlet-0 conditions in a bounded domain.The equation in this problem has concave and convex non-linear terms,and the non-linear term contains a sign-changing potential function.We use Nehari manifold to prove that there are at least two positive solutions of the equation(1)under some conditions.The other is the multiplicity of solutions for semi-linear elliptic problem which similar to the problem(1).In this paper,we prove that there are infinite solutions of the problem in a bounded domain.