An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P (lambda)) -Delta u = lambda b(x)|u| (q-2) u + a(x)|u| (p-2) u in Omega, a,u/a,n = 0 on a,Omega, where Omega is a bounded smooth domain in R (N) (N ae<yen> 2), 1 < q < 2 < p, lambda a R, and a, b a with 0 < alpha < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |lambda|. We then characterize the asymptotic profiles of these solutions as lambda -> 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P (lambda)).