Multiple positive solutions for semilinear elliptic equations in N involving concave-convex nonlinearities and sign-changing weight functions

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation - δu + u = λa (x) | u | ^{q - 2} u + b (x) | u | ^{p - 2} u in ℝ^N, u ∈ H¹ (ℝ^N), where λ < 0, 1 > q > 2 > p > 2* (2* = 2 N / (N - 2) if N ≥ 3, 2* = ∞ if N = 1,2), a (x), b (x) satisfy suitable conditions, and a (x) may change sign in N.