Lack of coercivity in a concave–convex type equation

Abstract

Existence and multiplicity of non-negative solutions are investigated for the concave–convex type equation $$-\Delta_p u +V(x)u^{p-1}=\lambda a(x) u^{r-1}+b(x)u^{q-1},\quad u\in W_0^{1,p}(\Omega),$$ where Ω is a bounded domain and 1 < r < p < q < p*. By minimization on the Nehari manifold, we find conditions on V, a, and b that yield up to four non-negative solutions when the left-hand side of the equation has a non-coercive behavior, a and b are sign-changing, and λ is positive and sufficiently small.