New multiplicity of positive solutions for some class of nonlocal problems

In this paper, we study the following nonlocal problem: {-(a - b integral Omega vertical bar del vertical bar(2) dx)Delta u = lambda vertical bar u vertical bar(q-2)u, x epsilon Omega, u = 0, x epsilon partial derivative Omega, where Omega is a smooth bounded domain in R-N with N >= 3, a, b > 0, 1 < q < 2 and lambda > 0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as b SE arrow 0.