An Indefinite and Critical Concave-Convex-Type Equation

We analyse the multiplicity of non-negative solutions for the critical concave-convex-type equation ?pu = ?m(x)up1 + ?a(x)ur1 + b(x)up*1, u W 1,p 0 (?),where ? is a bounded domain of RN, 1 < r < p and ? > 0, and ? is a real parameter. Combining minimization on the underlying Nehari manifold with energy estimates, we show that, under suitable conditions on a and b, three non-negative solutions may exist when ? is positive and sufficiently small and ? is in a right neighbourhood of ?1, the first weighted eigenvalue of the p-Laplacian. To the best of our knowledge, our multiplicity result is new, even in the semi-linear case p = 2. © Royal Society of Edinburgh 2013.