On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions

In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involvingweight functions, where the concave nonlinear termis lambda f (x)vertical bar u vertical bar(q-1) u and the convex nonlinear term is h(x)vertical bar u vertical bar(p-1) u with lambda is an element of R+. By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as lambda is an element of (0, lambda(*)), here the explicit expression of lambda(*) = lambda(*) (f, h, p, q, S) is provided.